A generalization of Krull-Webster's theory to higher order convex functions: multiple Γ -type functions
aa r X i v : . [ m a t h . C A ] S e p A generalization of Krull-Webster’s theory tohigher order convex functions:multiple Γ -type functions Jean-Luc Marichal (cid:3)
Naïm Zénaïdi y September 30, 2020
Abstract
We provide uniqueness and existence results for the eventually p -convex and eventually p -concave solutions to the difference equation ∆f = g on the open half-line ( ∞ ) , where p is a given nonnegative integer and g is a given function satisfying the asymptotic property that the sequence n ∆ p g ( n ) converges to zero. These solutions, that we call log Γ p -type functions, include various special functions such as the polygammafunctions, the logarithm of the Barnes G -function, and the Hurwitz zetafunction. Our results generalize to any nonnegative integer p the specialcase when p = obtained by Krull and Webster, who both generalizedBohr-Mollerup-Artin’s characterization of the gamma function.We also follow and generalize Webster’s approach and provide for log Γ p -type functions analogues of Euler’s infinite product, Weierstrass’infinite product, Gauss’ limit, Gauss’ multiplication formula, Legendre’sduplication formula, Euler’s constant, Stirling’s constant, Stirling’s for-mula, Wallis’s product formula, and Raabe’s formula for the gammafunction. We also introduce and discuss analogues of Binet’s function,Burnside’s formula, Fontana-Mascheroni’s series, Euler’s reflection for-mula, and Gauss’ digamma theorem.Lastly, we apply our results to several special functions, including theHurwitz zeta function and the generalized Stieltjes constants, and showthrough these examples how powerful is our theory to produce formulasand identities almost systematically. (cid:3) University of Luxembourg, Department of Mathematics, Maison du Nombre, 6, avenuede la Fonte, L-4364 Esch-sur-Alzette, Luxembourg. Email: [email protected] y University of Liège, Department of Mathematics, Allée de la Découverte, 12 - B37, B-4000Liège, Belgium. Email: [email protected] ey words and phrases. Difference equation, higher order convexity, Bohr-Mollerup-Artin’stheorem, Krull-Webster’s theory, generalized Stirling’s formula, generalized Stirling’s con-stant, generalized Euler’s constant, Euler’s reflection formula, Euler’s infinite product, Weier-strass’ infinite product, Gauss’ multiplication theorem, Gauss’ digamma theorem, Raabe’sformula, Wallis’s product formula, Fontana-Mascheroni’s series, Barnes G -function, Hurwitzzeta function, gamma-related function, multiple gamma-type function, generalized Stieltjesconstant. Contents g ( n ) is summable . . . . . . . . . . 183.3 Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 p -convex or p -concave functions . . . . . . . . . . . . 21 log Γ -type functions 25 Σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Multiple log Γ -type functions . . . . . . . . . . . . . . . . . . . . 265.3 Integration of multiple log Γ -type functions . . . . . . . . . . . . 28 log Γ -type functions 50 log Γ -type functions . . . . . . . . 517.2 Finding solutions from derivatives . . . . . . . . . . . . . . . . . 577.3 An alternative uniqueness result . . . . . . . . . . . . . . . . . . 622 Further results 63 log Γ -type functions . . . . . . . . . . . . 769.2 The gamma function . . . . . . . . . . . . . . . . . . . . . . . . . 779.3 The digamma and harmonic number functions . . . . . . . . . . 819.4 The polygamma functions . . . . . . . . . . . . . . . . . . . . . . 869.5 The Barnes G -function . . . . . . . . . . . . . . . . . . . . . . . . 959.6 The Hurwitz zeta function . . . . . . . . . . . . . . . . . . . . . . 1009.7 The generalized Stieltjes constants . . . . . . . . . . . . . . . . . 1069.8 Higher derivatives of the Hurwitz zeta function . . . . . . . . . . 1129.9 The principal indefinite sum of the Hurwitz zeta function . . . . 1179.10 The Catalan number function . . . . . . . . . . . . . . . . . . . . 119
10 Further examples 121
11 Conclusion 125A On Krull-Webster’s asymptotic condition 127B On a question raised by Webster 128C Asymptotic behaviors and bracketing 130 ist of symbols A Glaisher-Kinkelin’s constant b rn ( x ) D rx (cid:0) xn + r (cid:1) C k set of k times continuously differentiable functions on R + C k ( I ) set of k times continuously differentiable functions on ID ordinary derivative operator D p S { g : R + → R : ∆ p g ( t ) → as t → S ∞ }D ∞ S S p > D p S e D − N { g : R + → R : the sequence n g ( n ) is summable } f [ x
0, . . . , x p ] divided difference of f at the points x
0, . . . , x p f pn [ g ] function defined in (2) G n n th Gregory coefficient G n = R (cid:0) tn (cid:1) dtG n G n = − P nj = | G j | H x harmonic number function I arbitrary real interval whose interior is nonempty J q [ g ] Binet-like function defined in (38) K p K p + ∪ K p − K p + set of functions f : R + → R that are eventually p -convex K p − set of functions f : R + → R that are eventually p -concave K p ( I ) K p + ( I ) ∪ K p − ( I ) K p + ( I ) set of functions f : I → R that are p -convex K p − ( I ) set of functions f : I → R that are p -concave K ∞ T p > K p Log Γ p set of log Γ p -type functions (see Subsection 5.2) N , N ∗ N = {
0, 1, 2, . . . } , N ∗ = {
1, 2, . . . } P p [ f ] interpolating polynomial of degree p of f R + open half-line ( ∞ ) R p S { g : R + → R : for each x > , ρ pt [ g ]( x ) → as t → S ∞ } R p , m , n remainder in Gregory’s summation formula (44) ran ( Σ ) range of the map Σ S S = N or R x k x ( x − ) · · · ( x − k + ) x → S ∞ x tends to infinity, assuming only values in S ∈ { N , R } x + max { x } γ Euler’s constant γ [ g ] generalized Euler’s constant associated with gΓ p function Γ p / set of Γ p -type functions (see Subsection 5.2) ∆ forward difference operator ρ pa [ f ] function defined in (5) σ [ g ] , σ [ g ] σ [ g ] = R Σg ( t + ) dt , σ [ g ] = R Σg ( t ) dtΣ map defined in (23) 4 , ψ ν digamma function, polygamma functions Let R + denote the open half-line ( ∞ ) and let ∆ denote the forward differenceoperator on the space of functions from R + to R . In this paper, we are inter-ested in the classical functional equation ∆f = g on R + , which can be writtenexplicitly as f ( x + ) − f ( x ) = g ( x ) , x > where g : R + → R is a given function. This equation appears naturally in thetheory of the Euler gamma function, with f ( x ) = ln Γ ( x ) and g ( x ) = ln x , butalso in the study of many other special functions such as the Barnes G -functionand the Hurwitz zeta function.For any function g : R + → R , the difference equation ∆f = g has infinitelymany solutions, and each of them can be uniquely determined by prescribingits values in the interval (
0, 1 ] . Recall also that any two solutions differ by a -periodic function, i.e., a periodic function of period .For certain functions g , however, special solutions can be determined bytheir local properties or their asymptotic behaviors. On this issue, a seminalresult is the very nice characterization of the gamma function by Bohr andMollerup [20]. They showed that all log-convex solutions f : R + → R + to theequation f ( x + ) = x f ( x ) , x > (1)are of the form f ( x ) = c Γ ( x ) , where c > . Thus, the gamma function is akind of principal solution to its equation (Nörlund [69, Chapter 5] calls it the“Hauptlösung”). The additive, but equivalent, version of this result, obtainedby taking the logarithm of both sides of (1), can be stated as follows. For g ( x ) = ln x , all convex solutions f : R + → R to the difference equation ∆f = g are of the form f ( x ) = c + ln Γ ( x ) , where c ∈ R . Recall also that the proof of Bohrand Mollerup’s result was simplified later by Artin [9] (see also Artin [10]) and,as observed by Webster [80], this result “has then become known as the Bohr-Mollerup-Artin Theorem, and was adopted by Bourbaki [21] as the startingpoint for his exposition of the gamma function.”A noteworthy generalization of Bohr-Mollerup-Artin’s theorem was providedby Krull [46,47] and then independently by Webster [79,80]. Recall that a func-tion g : R + → R is said to be eventually convex (resp. eventually concave) ifit is convex (resp. concave) in a neighborhood of infinity. Krull [46] essentiallyshowed that for any eventually concave function g : R + → R having the asymp-totic property that, for each h > , g ( x + h ) − g ( x ) → as x → ∞ , f : R + → R to the equation ∆f = g (and dually, if g is eventually convex, then f is eventually concave). He also provided an explicit expression for this solutionas a pointwise limit of functions, namely f ( x ) = f ( ) + lim n → ∞ f n [ g ]( x ) , x > where f n [ g ]( x ) = − g ( x ) + n − X k = ( g ( k ) − g ( x + k )) + x g ( n ) . Much later, and independently, Webster [79, 80] established the multiplicativeversion of Krull’s result.In this paper, we generalize Krull-Webster’s result by relaxing the asymp-totic condition imposed on function g into the much weaker requirement thatthe sequence n ∆ p g ( n ) converges to zero for some nonnegative integer p .This relaxation leads us to replacing the convexity and concavity propertiesby the p -convexity and p -concavity properties (i.e., convexity and concavity oforder p ; see Definition 2.1 below). More precisely, we establish the unique-ness and existence theorems below (Theorems 1.1 and 1.2), as they were statedseparately by Webster in the case when p = .We let N denote the set of nonnegative integers and we let N ∗ denote the setof strictly positive integers. For any p ∈ N , any n ∈ N ∗ , and any g : R + → R ,we define the function f pn [ g ] : R + → R by the equation f pn [ g ]( x ) = − g ( x ) + n − X k = ( g ( k ) − g ( x + k )) + p X j = (cid:0) xj (cid:1) ∆ j − g ( n ) . (2) Theorem 1.1 (Uniqueness) . Let p ∈ N and let the function g : R + → R havethe property that the sequence n ∆ p g ( n ) converges to zero. Supposethat f : R + → R is an eventually p -convex or eventually p -concave functionsatisfying the difference equation ∆f = g . Then f is uniquely determined(up to an additive constant) by g through the equation f ( x ) = f ( ) + lim n → ∞ f pn [ g ]( x ) , x > Theorem 1.2 (Existence) . Let p ∈ N and suppose that the function g : R + → R is eventually p -convex or eventually p -concave and has the asymptoticproperty that the sequence n ∆ p g ( n ) converges to zero. Then there existsa unique (up to an additive constant) eventually p -convex or eventually p -concave solution f : R + → R to the difference equation ∆f = g . Moreover, f ( x ) = f ( ) + lim n → ∞ f pn [ g ]( x ) , x > (3) and f is p -convex (resp. p -concave) on any unbounded subinterval of R + on which g is p -concave (resp. p -convex).
6e observe that Theorem 1.2 was first proved in the case when p = by John [42]. As mentioned above, it was also established in the case when p = by Krull [46] and Webster [80]. More recently, the case when p = wasinvestigated by Rassias and Trif [72], but the asymptotic condition they imposedon function g is much stronger than ours and hence defines a very specificsubclass of functions. (We discuss Rassias and Trif’s result in Appendix A.)We also observe that attempts to establish Theorem 1.2 for any value of p weremade by Kuczma [50, Theorem 1] (see also Kuczma [52, pp. 118–121]) andthen by Ardjomande [8]. However, the representation formulas they providefor the solutions are rather intricate. Thus, to the best of our knowledge, bothTheorems 1.1 and 1.2, as stated above in their full generality and simplicity,were previously unknown.For any solution f arising from Theorem 1.2 when p = , Webster [80] callsthe function exp ◦ f a Γ -type function . In fact, exp ◦ f reduces to the gammafunction Γ when exp ◦ g is the identity function, which simply means that thegamma function restricted to R + is itself a Γ -type function. In this particularcase, the limit given in (3) reduces to the following Gauss well-known limit forthe gamma function Γ ( x ) = lim n → ∞ n ! n x x ( x + ) · · · ( x + n ) . (4)Similarly, for any fixed p ∈ N and any solution f arising from Theorem 1.2,we call the function exp ◦ f a Γ p -type function , and we naturally call the function f a log Γ p -type function . When the value of p is not specified, we call these func-tions multiple Γ -type function and multiple log Γ -type function , respectively.This terminology will be defined more formally and justified in Subsection 5.2.Interestingly, Webster established for Γ -type functions analogues of Leg-endre’s duplication formula , Gauss’ multiplication formula , Stirling’s for-mula , Euler’s constant , and
Weierstrass’ infinite product for the gammafunction. In this paper, we also establish for multiple Γ -type functions and mul-tiple log Γ -type functions analogues of all the formulas above as well as analoguesof Stirling’s constant , Euler’s infinite product , Wallis’s product formula , and
Raabe’s formula for the gamma function. We also introduce analogues of
Bi-net’s function , Burnside’s formula , and
Fontana-Mascheroni’s series , anddiscuss analogues of
Euler’s reflection formula and
Gauss’ digamma theo-rem . Thus, for each multiple Γ -type function, it is no longer surprising forinstance that an analogue of Euler’ infinite product must hold, almost render-ing a formal proof unnecessary! All these results, together with the uniquenessand existence theorems above, show that our theory provides a very general andunified framework to study the properties of a large variety of functions. Thus,for each of these functions we can retrieve known formulas and establish newones. 7 xample 1.3 (The Hurwitz zeta function, see Subsection 9.6) . Consider theHurwitz zeta function s ζ ( s , a ) , defined when ℜ ( a ) > as an analyticcontinuation to C \ { } of the series P ∞ k = ( a + k ) − s . This function is known tosatisfy the difference equation ζ ( s , a + ) − ζ ( s , a ) = − a − s . Also, it is not difficult to see that, for any s ∈ R \ { } , the restriction of the map x ζ ( s , x ) to R + is a log Γ p ( s ) -type function, where p ( s ) = max { ⌊ − s ⌋ } . Theorem 1.2 then tells us that all eventually p ( s ) -convex or eventually p ( s ) -concave solutions f s : R + → R to the difference equation f s ( x + ) − f s ( x ) = − x − s are of the form f s ( x ) = c s + ζ ( s , x ) , where c s ∈ R . Moreover, equation (3)provides the following analogue of Gauss’ limit for the gamma function ζ ( s , x ) = ζ ( s ) + x − s + lim n → ∞ n − X k = (cid:0) ( x + k ) − s − k − s (cid:1) − p ( s ) X j = (cid:0) xj (cid:1) ∆ j − n n − s , where s ζ ( s ) is the Riemann zeta function. Using one of our new results(namely, Theorem 6.5), we are also able to derive the following analogue of Stirling’s formula ζ ( s , x ) − x − s s − − p ( s ) X j = G j ∆ j − x x − s → as x → ∞ , where G n = R (cid:0) tn (cid:1) dt is the n th Gregory coefficient. For instance, setting s = − in this asymptotic formula, we obtain ζ (cid:0) −
32 , x (cid:1) + x / − x / + ( x + ) / → as x → ∞ . Example 1.4 (Barnes’s G -function, see Subsection 9.5) . The Barnes G -function G : R + → R + is the unique solution to the equation f ( x + ) = Γ ( x ) f ( x ) whose logarithm is eventually -convex and vanishes at x = . Thus defined, thisfunction is a Γ -type function. In particular, formula (3) provides the followinganalogue of Gauss’ limit for the gamma function G ( x ) = lim n → ∞ Γ ( ) Γ ( ) · · · Γ ( n ) Γ ( x ) Γ ( x + ) · · · Γ ( x + n ) n ! x n ( x ) . Euler’s infinite product G ( x ) = Γ ( x ) ∞ Y k = Γ ( k ) Γ ( x + k ) k x ( + /k )( x ) and the following analogue of Weierstrass’ infinite product G ( x ) = e (− γ − ) ( x ) Γ ( x ) ∞ Y k = Γ ( k ) Γ ( x + k ) k x e ψ ′ ( k ) ( x ) , where γ is Euler’s constant and ψ is the digamma function. We also have thefollowing surprising analogues of Wallis’s product formula lim n → ∞ Γ ( ) Γ ( ) · · · Γ ( n − ) Γ ( ) Γ ( ) · · · Γ ( n ) (cid:18) ne (cid:19) n = √ and lim n → ∞ G ( ) G ( ) · · · G ( n − ) G ( ) G ( ) · · · G ( n ) n n − n −
124 2 n − π n e n − n − = A
12 , where A is Glaisher-Kinkelin’s constant defined by the equation ζ ′ (− ) = − ln A .Throughout this paper we will use the basic function g ( x ) = ln x as theguiding example. However, many other functions, including the examples above,will be discussed in Section 9.This paper is outlined as follows. In Section 2, we present some defini-tions and preliminary results on higher order convexities as well as on Newtoninterpolation theory. In Section 3, we establish Theorems 1.1 and 1.2 and pro-vide conditions for the sequence n f pn [ g ]( x ) to converge uniformly on anybounded subset of R + . We also examine the particular case when the sequence n g ( n ) is summable, and we provide historical remarks on some improve-ments of Krull-Webster’s theory. In Section 4, we investigate some propertiesof the set of functions g ( x ) defined by the asymptotic condition stated in The-orems 1.1 and 1.2. We also investigate the subset of those functions that areeventually p -convex or eventually p -concave. In Section 5, we introduce, in-vestigate, and characterize the multiple log Γ -type functions. In Section 6, weshow how Stirling’s formula, Stirling’s constant, and Euler’s constant can begeneralized to the multiple log Γ -type functions and we introduce analogues ofBinet’s function, Burnside’s formula, and Fontana-Mascheroni’s series. We alsoshow how the so-called Gregory summation formula, with an integral form ofthe remainder, can be very easily derived in this setting. In Section 7, we discussconditions for the solutions arising from Theorem 1.2 (i.e., the log Γ p -type func-tions) to be differentiable and we show how these solutions can also be obtained9y first differentiating both sides of the difference equation ∆f = g . In Section 8,we explore further properties of the multiple log Γ -type functions. Specifically,we provide analogues of Euler’s infinite product, Weierstrass’ infinite product,Raabe’s formula, Gauss’ multiplication formula, and Wallis’s product formula.We also discuss analogues of Euler’s reflection formula and Gauss’ digammatheorem, and we define and solve a generalized version of a functional equationproposed by Webster. In Sections 9 and 10, we apply our results to a numberof multiple Γ -type functions and multiple log Γ -type functions, many of whoseare well-known special functions related to the gamma function.We use the following notation throughout. The symbol I denotes an (ar-bitrary) interval of the real line whose interior is nonempty. For any points x x
1, . . . , x p + ∈ I and any function f : I → R , the symbol f [ x x
1, . . . , x p + ] stands for the divided difference of f at the points x x
1, . . . , x p + . The symbol S represents either N or R . For any S ∈ { N , R } , the notation x → S ∞ meansthat x tends to infinity, assuming only values in S . For any x ∈ R and any k ∈ N , we set x + = max { x } and x k = x ( x − ) · · · ( x − k + ) = Γ ( x + ) Γ ( x − k + ) . For any k ∈ N and any nonempty open real interval I , we let C k ( I ) denote the setof k times continuously differentiable functions on I , and we set C k = C k ( R + ) .We also let ∆ and D denote the usual difference and derivative operators, re-spectively. We sometimes add a subscript to specify the variable on which theoperator acts, e.g., writing ∆ n and D x .Recall that the digamma function ψ is defined on R + by the equation ψ ( x ) = D ln Γ ( x ) . The polygamma functions ψ ν ( ν ∈ Z ) are defined on R + as follows.If ν ∈ N , then ψ ν ( x ) = D ν ψ ( x ) . In particular, ψ = ψ is the digamma function.If ν ∈ Z \ N , then we have ψ − ( x ) = ln Γ ( x ) and ψ ν − ( x ) = Z x ψ ν ( t ) dt = Z x ( x − t ) − ν − (− ν − ) ! ln Γ ( t ) dt . Recall also that the harmonic number function x H x is defined on (− ∞ ) by the series H x = ∞ X k = (cid:18) k − x + k (cid:19) . Both functions are strongly related: we have H x − = ψ ( x ) + γ on R + , where γ is Euler’s constant (also called Euler-Mascheroni constant).For any a > , any p ∈ N , and any g : R + → R , we define the function ρ pa [ g ] : [ ∞ ) → R by the equation ρ pa [ g ]( x ) = g ( x + a ) − p − X j = (cid:0) xj (cid:1) ∆ j g ( a ) . (5)10or any p ∈ N and any S ∈ { N , R } , we let R p S be the set of functions g : R + → R having the asymptotic property that, for each x > , ρ pt [ g ]( x ) → as t → S ∞ . We also let D p S be the set of functions g : R + → R having the asymptoticproperty that ∆ p g ( t ) → as t → S ∞ . We immediately observe that the inclusion D p S ⊂ D p + holds for every p ∈ N .We will see in Subsection 4.1 that so does the inclusion R p S ⊂ R p + . This section is devoted to some basic definitions and results that are needed inthis paper. We essentially focus on higher order convexity and Newton inter-polation theory.
Let us recall the definition of higher order convexity and concavity propertiesand present some related results. For background see, e.g., [50], [53, Chapter 15],[70], and [73, pp. 237–240].
Definition 2.1.
A function f : I → R is said to be convex of order p or simply p -convex for some integer p > − if for any system x < x < · · · < x p + of p + points in I it holds that f [ x x
1, . . . , x p + ] > The function f is said to be concave of order p or simply p -concave if − f is p -convex.Thus defined, a function f : I → R is -convex (resp. -concave) if it is anordinary convex (resp. concave) function, while it is a -convex (resp. -concave)if it is an increasing (resp. decreasing) function.For any integer p > − , we let K p + ( I ) (resp. K p − ( I ) ) denote the set of p -convex (resp. p -concave) functions f : I → R and we let K p + (resp. K p − ) denotethe set of functions f : R + → R that are eventually p -convex (resp. eventually p -concave), i.e., p -convex (resp. p -concave) in a neighborhood of infinity. Wealso set K p ( I ) = K p + ( I ) ∪ K p − ( I ) and K p = K p + ∪ K p − . The following lemma provides some known connections between higher orderconvexity and higher order differentiability (see, e.g., [53, Chapter 15]).11 emma 2.2.
Suppose that I is an nonempty open real interval and let p ∈ N ∗ . Then the following assertions hold.(a) We have K p ( I ) ⊂ C p − ( I ) .(b) If f ∈ K p + ( I ) , then ∆ j f ∈ K p − j + ( I ) for j =
0, . . . , p + .(c) If f ∈ C j ( I ) ∩ K p + ( I ) for some j ∈ {
0, . . . , p + } , then f ( j ) ∈ K p − j + ( I ) .(d) If f ∈ C p ( I ) , then f ∈ K p + ( I ) if and only if f ( p ) ∈ K + ( I ) .(e) If f ∈ C p ( I ) , then f ∈ K p − + ( I ) if and only if f ( p ) ∈ K − + ( I ) .(f) We have f ∈ K p + ( I ) if and only if f ∈ C p − ( I ) and f ( p − ) ∈ K + ( I ) .(g) If f ∈ C ( I ) and f ′ ∈ K p − + ( I ) , then f ∈ K p + ( I ) . We also have the following important lemma. It is interesting in its ownright and will be very useful in the subsequent sections. A variant of this resultcan be found in Kuczma [53, Lemma 15.7.2]. Recall first that for any f : I → R ,any p ∈ N , and any x ∈ I such that x + p ∈ I , we have ∆ p f ( x ) = p ! f [ x , x +
1, . . . , x + p ] , (6)where ∆ stands for the standard forward difference operator. Lemma 2.3.
Let p ∈ N . A function f : I → R is p -convex (resp. p -concave)if and only if the map ( z
0, . . . , z p ) ∈ I p + f [ z
0, . . . , z p ] is increasing (resp.decreasing) in each place. In particular, if f is p -convex (resp. p -concave)and if ∆ p f is defined on I , then ∆ p f is increasing (resp. decreasing) on I .Proof. Using the definition of p -convexity and the standard recurrence relationfor divided differences, we can see that f is p -convex if and only if, for anypairwise distinct x
0, . . . , x p ∈ I , we have f [ x x x p ] − f [ x x x p ] x − x > Equivalently, for any pairwise distinct x
0, . . . , x p ∈ I , we have x > x ⇒ f [ x x x p ] − f [ x x x p ] > The latter condition exactly means that the map ( z
0, . . . , z p ) f [ z
0, . . . , z p ] isincreasing in the first place. Since this map is known to be symmetric, it mustbe increasing in each place. The second part of the lemma follows from (6).12 .2 Newton interpolation For any integer p ∈ N , any p points x
0, . . . , x p − ∈ R + , and any function f : R + → R , we let the map x P p − [ f ]( x
0, . . . , x p − x ) denote the unique interpolating polynomial of f with nodes at x
0, . . . , x p − .Recall that this polynomial has degree at most p − . (The zero polynomialcan be assumed to have degree − .) For instance, using the classical Newtoninterpolation formula we obtain the following identity: for any a > , P p − [ f ]( a , a +
1, . . . , a + p − x ) = p − X j = (cid:0) x − aj (cid:1) ∆ j f ( a ) . (7)Also, the corresponding interpolation error at x is f ( x ) − p − X j = (cid:0) x − aj (cid:1) ∆ j f ( a ) = ( x − a ) p f [ a , a +
1, . . . , a + p − x ] (8)(see, e.g., [71, Section 8.2.2] and [77, Section 2.1.3]). The right side of (8) isactually the remainder of the ( p − ) th degree Newton expansion of f ( x ) about x = a (see, e.g., [34, Section 5.3]). Note also that formula (8), which actuallygeneralizes (6) on R + , is a pure identity and is therefore valid without anyrestriction on the form of f ( x ) . When f ∈ C p , the right side of (8) also takesthe form (cid:0) x − ap (cid:1) f ( p ) ( ξ ) for some real number ξ satisfying min { a , x } < ξ < max { a + p − x } . Using (7) and (8) we see that, for any a > , any p ∈ N , and any g : R + → R , the quantity ρ pa [ g ]( x ) defined in (5) is precisely the interpolation error at a + x when considering the interpolating polynomial of g with nodes at a , a +
1, . . . , a + p − . We then immediately derive the following identities: ρ pa [ g ]( x ) = g ( x + a ) − P p − [ g ]( a , a +
1, . . . , a + p − a + x ) , (9) ρ pa [ g ]( x ) = x p g [ a , a +
1, . . . , a + p − a + x ] . (10)We now provide a key technical lemma that will be used repeatedly in thispaper to obtain various convergence results. Lemma 2.4.
Let p ∈ N , f ∈ K p , and a > be so that f is p -convex or p -concave on [ a , ∞ ) . Then, for any x > , we have ± ε p ( x ) ρ p + a [ f ]( x ) ± (cid:12)(cid:12)(cid:12)(cid:0) x − p (cid:1)(cid:12)(cid:12)(cid:12) ⌈ x ⌉ − X j = ∆ p + f ( a + j ) , here ε p ( x ) ∈ { −
1, 0, 1 } is the sign of x p + and ± stands for or − ac-cording to whether f ∈ K p + or f ∈ K p − . Moreover, if x ∈ {
0, 1, . . . , p } (i.e. ε p ( x ) = ), then ρ p + a [ f ]( x ) = .Proof. Let us first establish the inequalities. Negating f if necessary, we mayassume that it lies in K p + . We may also assume that x / ∈ {
0, 1, . . . , p } , whichmeans that ε p ( x ) = . By (10) we have ε p ( x ) ρ p + a [ f ]( x ) = ε p ( x ) x p + f [ a , a +
1, . . . , a + p , a + x ] > and hence, using Lemma 2.3, identity (6), and the standard recurrence relationfor divided differences, we obtain ε p ( x ) ρ p + a [ f ]( x )= ε p ( x ) x p + f [ a , a +
1, . . . , a + p , a + x ]= ε p ( x )( x − ) p ( f [ a + x , a +
1, . . . , a + p ] − f [ a , a +
1, . . . , a + p ]) ε p ( x ) (cid:0) x − p (cid:1) ( ∆ p f ( a + x ) − ∆ p f ( a )) ε p ( x ) (cid:0) x − p (cid:1) ( ∆ p f ( a + ⌈ x ⌉ ) − ∆ p f ( a )) , which proves the inequalities. Now, when x ∈ {
0, 1, . . . , p } , then (cid:0) xj (cid:1) = when-ever j > x and hence ρ p + a [ g ]( x ) = by (5). Remark . It is not difficult to see that, in Lemma 2.4, the upper delimiter ⌈ x ⌉ − of the sum could be replaced with ⌈ median { x , 1, x − p + } ⌉ − whenever p > . Although this alternative delimiter would make the second inequality alittle tighter, it would not have a great impact on our subsequent results. In this section we establish Theorems 1.1 and 1.2 and show that, under theassumptions of these theorems, the sequence n f pn [ g ]( x ) converges uniformlyon any bounded subset of R + . We also discuss the particular case where thesequence n g ( n ) is summable. Lastly, we provide historical notes on Krull-Webster’s theory and some of its improvements. We start this section by establishing a slightly improved version of our unique-ness Theorem 1.1. We state this new version in Theorem 3.1 below and providea very short proof. Let us first note that any solution f : R + → R to the equation14 f = g satisfies the equations f ( n ) = f ( ) + n − X k = g ( k ) , n ∈ N ∗ ; (11) f ( x + n ) = f ( x ) + n − X k = g ( x + k ) , n ∈ N ∗ . (12)Also, using (2), (5), (11), and (12), we can easily derive the identity f ( x ) = f ( ) + f pn [ g ]( x ) + ρ p + n [ f ]( x ) . (13) Theorem 3.1 (Uniqueness) . Let p ∈ N and g ∈ D p S . Suppose that f : R + → R is a solution to the equation ∆f = g that lies in K p . Then, the followingassertions hold.(a) We have f ∈ R p + .(b) For each x > , the sequence n f pn [ g ]( x ) converges and we have f ( x ) = f ( ) + lim n → ∞ f pn [ g ]( x ) , x > (c) For any nonempty bounded subset E of R + , the sequence n f pn [ g ] converges uniformly on E to f − f ( ) .Proof. We clearly have f ∈ D p + . Assertion (a) then follows from Lemma 2.4.Assertion (b) follows from assertion (a) and identity (13). Using again identity(13) and Lemma 2.4, for large integer n we obtain sup x ∈ E | f pn [ g ]( x ) − f ( x ) + f ( ) | = sup x ∈ E (cid:12)(cid:12) ρ p + n [ f ]( x ) (cid:12)(cid:12) sup x ∈ E (cid:12)(cid:12)(cid:12)(cid:0) x − p (cid:1)(cid:12)(cid:12)(cid:12) sup x ∈ E ⌈ x ⌉ − X j = (cid:12)(cid:12) ∆ p + f ( n + j ) (cid:12)(cid:12) . This establishes assertion (c).
Example 3.2.
Using Theorem 3.1 with g ( x ) = ln x and p = , it follows thatall solutions f : R + → R + to the equation f ( x + ) = x f ( x ) for which ln f liein K are of the form f ( x ) = c Γ ( x ) , where c > . We thus simply retrieveBohr-Mollerup-Artin’s theorem as expected, as well as Gauss’ limit (4).Using the definition of ρ pa [ g ]( x ) , we can easily derive the following two iden-tities: ρ pa [ g ]( p ) = ∆ p g ( a ) ; (14) ρ pa [ g ]( x ) − ρ p + a [ g ]( x ) = (cid:0) xp (cid:1) ρ pa [ g ]( p ) . (15)15dentity (14) shows that the inclusion R p S ⊂ D p S holds for any p ∈ N . We will seein Subsection 4.1 that the converse inclusion does not hold. Now, the followingstraightforward identities will also be useful as we continue: f pn + [ g ]( x ) − f pn [ g ]( x ) = − ρ p + n [ g ]( x ) ; (16) f pn [ g ]( x + ) − f pn [ g ]( x ) = g ( x ) − ρ pn [ g ]( x ) . (17)For any integers m n , from (16) we obtain f pn [ g ]( x ) = f pm [ g ]( x ) − n − X k = m ρ p + k [ g ]( x ) , (18)which shows that, for any x > , the convergence of the sequence n f pn [ g ]( x ) is equivalent to the summability of the sequence n ρ p + n [ g ]( x ) .We now establish a slightly improved version of our existence Theorem 1.2.We first present a technical lemma, which follows straightforwardly from Lemma2.4. Lemma 3.3.
Let p ∈ N , g ∈ K p , and m ∈ N ∗ be so that g is p -convex or p -concave on [ m , ∞ ) . Then, for any x > and any integer n > m , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − X k = m ρ p + k [ g ]( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:0) x − p (cid:1)(cid:12)(cid:12)(cid:12) ⌈ x ⌉ − X j = | ∆ p g ( n + j ) − ∆ p g ( m + j ) | . Theorem 3.4 (Existence) . Let p ∈ N and g ∈ D p S ∩ K p . Then, the followingassertions hold.(a) We have g ∈ R p S .(b) For each x > , the sequence n f pn [ g ]( x ) converges and the function f : R + → R defined by f ( x ) = lim n → ∞ f pn [ g ]( x ) , x > is a solution to the equation ∆f = g that is p -concave (resp. p -convex)on any unbounded subinterval I of R + on which g is p -convex (resp. p -concave). Moreover, we have f ( ) = and, for every n ∈ I ∩ N ∗ , | f pn [ g ]( x ) − f ( x ) | ⌈ x ⌉ (cid:12)(cid:12)(cid:12)(cid:0) x − p (cid:1)(cid:12)(cid:12)(cid:12) | ∆ p g ( n ) | , x > (c) For any nonempty bounded subset E of R + , the sequence n f pn [ g ] converges uniformly on E to f . roof. We have g ∈ D p S ⊂ D p + . By Lemma 2.4, it follows immediately that g lies in R p + , and hence also in R p S by (14) and (15). This establishes assertion(a). Now, suppose for instance that g lies in K p + . Let I be any unboundedsubinterval of R + on which g is p -convex and let m ∈ I ∩ N ∗ . For any x > ,the sequence k ρ p + k [ g ]( x ) for k > m does not change in sign by Lemma 2.4.Thus, since g lies in D p N , for any x > the sequence n n − X k = m ρ p + k [ g ]( x ) converges by Lemma 3.3. By (18) it follows that the sequence n f pn [ g ]( x ) converges. Denoting the limiting function by f , by (17) and assertion (a) wemust have ∆f = g . Moreover, we also have f ( ) = by Theorem 3.1.It is also easy to see that every f pn [ g ] is p -concave on I . (Note that thesecond sum in (2) is a polynomial of degree p in x , hence it is both p -convexand p -concave.) Since f is a pointwise limit of functions p -concave on I , it toois p -concave on I .The inequality then follows from Eq. (13), Lemma 2.4, and the observationthat the restriction of the sequence n ∆ p g ( n ) to I ∩ N ∗ increases to zero byLemma 2.3. This proves assertion (b). Assertion (c) immediately follows fromassertion (b).Theorems 3.1 and 3.4 show that the assumption g ∈ D p N ∩ K p constitutes asufficient condition to ensure both the uniqueness (up to an additive constant)and existence of solutions to the equation ∆f = g that lie in K p . Nevertheless,we can show that this condition is actually not quite necessary. We discuss andelaborate on this natural question in Appendix B.We now present an important property of the sequence n f pn [ g ]( x ) . Con-sidering the straightforward identity f p + n [ g ]( x ) − f pn [ g ]( x ) = (cid:0) xp + (cid:1) ∆ p g ( n ) , we immediately see that if the sequence n f p + n [ g ]( x ) − f pn [ g ]( x ) approacheszero for some x ∈ R + \ {
0, 1, . . . , p } , then necessarily g ∈ D p N . More importantly,this identity also shows that if g ∈ D p N and if the sequence n f pn [ g ]( x ) con-verges, then so does the sequence n f p + n [ g ]( x ) and both sequences convergeto the same limit. Since we have D p N ⊂ D p + N for any p ∈ N , we immediatelyobtain the following important proposition. Proposition 3.5.
Let p ∈ N . If g ∈ D p N and if the sequence n f pn [ g ]( x ) converges, then for any integer q > p the sequence n | f pn [ g ]( x ) − f qn [ g ]( x ) | converges to zero. Moreover, the convergence is uniform on any nonemptybounded subset of R + . .2 The case when the sequence g ( n ) is summable Let e D − N be the set of functions g : R + → R having the asymptotic propertythat the sequence n P n − k = g ( k ) converges. We immediately observe that e D − N ⊂ D N . In this context, our uniqueness and existence results reduce to thefollowing two theorems. Theorem 3.6 (Uniqueness) . Let g ∈ e D − N and suppose that f : R + → R isa solution to the equation ∆f = g that lies in K . Then, the followingassertions hold.(a) f ( x ) has a finite limit as x → ∞ , denote it by f ( ∞ ) .(b) For each x > , the sequence n P n − k = g ( x + k ) converges and wehave f ( x ) = f ( ∞ ) − ∞ X k = g ( x + k ) , x > (c) The sequence n P n − k = g ( x + k ) converges uniformly on R + to f ( ∞ )− f ( x ) .Proof. The sequence n f ( n ) converges by (11). Assuming for instance that f ∈ K + , for any x > we obtain f ( ⌊ x ⌋ + n ) f ( x + n ) f ( ⌈ x ⌉ + n ) for large integer n . Letting n → N ∞ in these inequalities and using the squeeze theorem, we getassertion (a). Assertion (b) follows from assertion (a) and identity (12). Now,for large integer n , by (12) we have sup x ∈ R + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k = n g ( x + k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sup x ∈ R + | f ( x + n ) − f ( ∞ ) | | f ( n ) − f ( ∞ ) | . This proves assertion (c).
Theorem 3.7 (Existence) . Let g ∈ e D − N ∩ K . Then, the following assertionshold.(a) We have g ∈ R N .(b) For each x > , the sequence n P n − k = g ( x + k ) converges and thefunction f : R + → R defined by f ( x ) = − ∞ X k = g ( x + k ) , x > s a solution to the equation ∆f = g that is decreasing (resp. increas-ing) on any unbounded subinterval I of R + on which g is increas-ing (resp. decreasing). Moreover, we have f ( ∞ ) = and, for every n ∈ I ∩ N ∗ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k = n g ( x + k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | f ( n ) | , x > (c) The sequence n P n − k = g ( x + k ) converges uniformly on R + to − f ( x ) .Proof. This follows straightforwardly from Theorems 3.4 and 3.6.
As mentioned in the Introduction, the uniqueness and existence result in thecase when p = was established in the pioneering work of Krull [46,47] and thenindependently by Webster [79, 80] as a generalization of Bohr-Mollerup-Artin’stheorem. We observe that it was also partially rediscovered by Dinghas [29].In addition, we note that Krull’s result was presented and somewhat revisitedby Kuczma [48] (see also Kuczma [51] and Kuczma [52, pp. 114–118]) as wellas by Anastassiadis [6, pp. 69–73]. To our knowledge, the only attempts toestablish uniqueness and existence results for any value of p were made byKuczma [52, pp. 118–121] and Ardjomande [8]. Independently of these latterresults, a special investigation of the case when p = , which involves the Barnes G -function, was made by Rassias and Trif [72] (see our Appendix A).We also observe that Gronau and Matkowski [37, 38] improved the multi-plicative version of Krull’s result by replacing the log-convexity property withthe much weaker condition of geometrical convexity (see also Guan [39] for arecent application of this result), thus providing another characterization ofthe gamma function, which was later improved by Alzer and Matkowski [3]and Matkowski [58]. (For further characterizations of the gamma function andgeneralizations, see also Anastassiadis [6] and Muldoon [67].)Many other variants and improvements of Krull’s result can actually befound in the literature. For instance, Anastassiadis [5] (see also [6, p. 71]) gen-eralized Krull’s result by modifying the asymptotic condition. Rohde [74] alsogeneralized that result by modifying the convexity property. Gronau [35] pro-posed a variant of Krull’s result and applied it to characterize the Euler beta andgamma functions and study certain spirals (see also Gronau [36]). Merkle andRibeiro Merkle [60] proposed to combine Krull’s result with differentiation tech-niques to characterize the Barnes G -function. Himmel and Matkowski [41] alsoproposed improvements of Krull’s result to characterize the beta and gammafunctions. 19 Interpretations of the asymptotic conditions
In this section, we provide interpretations of the asymptotic conditions thatdefine the sets R p S and D p S and we investigate some properties of these sets.We also describe the sets R p S ∩ K p and D p S ∩ K p and show that they actuallycoincide. Moreover, we show that C p ∩ D p S ∩ K p is exactly the set of functions g ∈ C p for which g ( p ) eventually increases or decreases to zero. Using (9) and (10), we can immediately state the following characterization ofthe set R p S in terms of interpolating polynomials. Using (9), (10), and (14), wecan obtain a similar characterization for the set D p S . Proposition 4.1.
Let p ∈ N . A function g : R + → R lies in R p S if and onlyif, for each x > , g [ a , a +
1, . . . , a + p − a + x ] → as a → S ∞ . (19) When S = R (resp. S = N ), condition (19) means that g asymptoticallycoincides with its interpolating polynomial whose nodes are any p pointsequally spaced by (resp. any p consecutive integers). It is clear that the sets R p S and D p S are closed under linear combinations;hence they are linear spaces. Moreover, using (14) and (15) we see that R p S = R p + ∩ D p S . (20)In particular, the sets R R are increasingly nested. As already observed,this property also holds trivially for the sets D D . Now, identity (9)shows that the polynomial function x x p lies in R p + \ R p S and we can easilysee that it lies also in D p + \ D p S . Thus, we have proved that R p S R p + and D p S D p +
1S .
We also have R p S D p S for any p ∈ N ∗ . Indeed, the -periodic function g ( x ) = sin ( πx ) lies in D p S \ R p S for any p ∈ N ∗ . On the other hand, we have R R = D R R N D N . For instance, we can easily construct a continuous (or even smooth) function g : R + → R such that for any n ∈ N ∗ , we have g = on the interval [ n − n − n ] and g ( n − n ) = . Such a function has the property that, for each x > , g ( x + n ) → as n → N ∞ . However, since it does not vanish at infinity, it mustlie in R N \ R R . 20t is clear that if a function f : R + → R lies in D p + , then ∆f lies in D p S .Also, if f lies in R p + , then ∆f lies in R p S by (20). This latter observation followsalso from the second of the straightforward identities ρ pa + [ f ]( x ) − ρ pa [ f ]( x ) = ρ pa [ ∆f ]( x ) ; (21) ρ p + a [ f ]( x + ) − ρ p + a [ f ]( x ) = ρ pa [ ∆f ]( x ) . (22)Thus, we have proved the following proposition. Proposition 4.2.
Let j , p ∈ N be such that j p . Then the followingassertions hold.(a) If f ∈ R p S , then ∆ j f ∈ R p − j S .(b) f ∈ D p S if and only if ∆ j f ∈ D p − j S . We will see in Corollary 4.7 that, if f ∈ K p − , then the implication inassertion (a) of Proposition 4.2 becomes an equivalence.It is easy to see that, for any p ∈ N , the space R p S contains every functionthat behaves asymptotically like a polynomial of degree less than or equal to p − ; that is, every function g : R + → R such that g ( x ) − P ( x ) → as x → ∞ for some polynomial P of degree less than or equal to p − . More generally, if g − h ∈ R p S and h ∈ R p S , then clearly g ∈ R p S . To give another illustration of thislatter property, we observe for instance that both functions ln x and H x − ln x lie in R R and hence so does the function H x (which, a priori, is a not completelytrivial result).It is clear that the spaces R ∞ S = ∪ p > R p S and D ∞ S = ∪ p > D p S contain avery large variety of functions, including not only all the functions that havepolynomial behaviors at infinity as discussed above, and in particular all therational functions, but also many other functions. We observe, however, thatthey do not contain any strictly increasing exponential function. For instance,if g ( x ) = x , then ∆ p g ( x ) = x for any p ∈ N , and this function does notvanish at infinity. Actually, such exponential functions grow asymptoticallymuch faster than polynomial functions and may remain eventually p -convexeven after adding nonconstant -periodic functions. For instance, both functions x and x + sin ( πx ) are eventually p -convex for any p ∈ N . Remark . Using (5) and (20), we also obtain R p S = R ∞ S ∩ D p S for any p ∈ N . p -convex or p -concave functions Let us now investigate the sets K p , R p S ∩ K p , and D p S ∩ K p . It is easy to see thatnone of these sets is a linear space. For instance, both functions f ( x ) = x + sin x and g ( x ) = x lie in K but f − g does not. We also have ∆f / ∈ K , whichshows that K p is not closed under the operator ∆ . Finally, we can see that21oth functions f ( x ) = x + sin xx and g ( x ) = x lie in R ∩ K (use, e.g.,Proposition 4.5 and Theorem 4.9 below) but f − g does not.The following proposition shows that, just as the sets C C C
2, . . . are de-creasingly nested, so are the sets K − K K
1, . . . and thus we can naturally let K ∞ denote the intersection set ∩ p > K p . Proposition 4.4.
For any integer p > − , we have K p + K p .Proof. Let f ∈ K p + for some integer p > − . Suppose without loss ofgenerality that f ∈ K p + + and let I be an unbounded subinterval of R + onwhich f is ( p + ) -convex. By Lemma 2.3, it follows that the restriction ofthe map ( z
0, . . . , z p + ) f [ z
0, . . . , z p + ] to I p + is increasing in each place.If f / ∈ K p − , then there are pairwise distinct points x
0, . . . , x p + ∈ I such that f [ x
0, . . . , x p + ] > . But then, f is p -convex on the interval ( max i x i , ∞ ) , thatis, f ∈ K p + . To see that the strict inclusion holds, we just observe that thefunction f ( x ) = x p + + sin x lies in K p \ K p + .Interestingly, Proposition 4.4 shows that the assumption g ∈ K p , whichoccurs in many statements (e.g., in Theorem 3.4), can be given equivalently by g ∈ ∪ q > p K q .We also have the following two important propositions. Proposition 4.5.
For any p ∈ N , we have R p R ∩ K p = D p R ∩ K p = R p N ∩ K p = D p N ∩ K p .Proof. We already know that R p S ⊂ D p S and D p R ⊂ D p N . Also, D p S ∩ K p ⊂ R p S by Theorem 3.4. It remains to show that D p N ∩ K p ⊂ D p R . Let g ∈ D p N ∩ K p .Suppose for instance that g ∈ K p + and let a > be so that g is p -convex on [ a , ∞ ) . By Lemma 2.3, ∆ p g is increasing on [ a , ∞ ) . Thus, for any x > a + ,we have ∆ p g ( ⌊ x ⌋ ) ∆ p g ( x ) ∆ p g ( ⌈ x ⌉ ) . Letting x → ∞ and using the squeeze theorem, we obtain that g ∈ D p R . Proposition 4.6. If f ∈ K p for some p ∈ N , then the following assertionsare equivalent: (i) f ∈ R p +
1S , (ii) f ∈ D p +
1S , (iii) ∆f ∈ R p S , (iv) ∆f ∈ D p S .
Proof.
We clearly have (i) ⇒ (ii) and (iii) ⇒ (iv). By Proposition 4.2, we alsohave (i) ⇒ (iii) and (ii) ⇒ (iv). Finally, by Theorem 3.1, we have (iv) ⇒ (i).Combining Lemma 2.2(b) with Propositions 4.2 and 4.6, we obtain the fol-lowing corollary, which naturally complements Proposition 4.2. Corollary 4.7.
Let j , p ∈ N be such that j p . If f ∈ K p − , then we have f ∈ R p S if and only if ∆ j f ∈ R p − j S . D p ∩ K p instead of D p S ∩ K p .We also observe that when g lies in D p ∩ K p , then by (10) and Lemma 2.3the maps t ρ pt [ g ]( x ) and t ∆ p g ( t ) eventually increase or decrease tozero. It is not known whether these latter conditions characterize the set D p ∩ K p . However, when g lies in C p , we have the nice characterization given inTheorem 4.9 below, which immediately follows from the next proposition. Proposition 4.8.
Let p , r ∈ N be such that r p and let g ∈ C r . Then thefollowing assertions hold.(a) g ∈ K p + if and only if g ( r ) ∈ K p − r + . More precisely, for any unboundedopen interval I of R + , g is p -convex on I if and only if g ( r ) is ( p − r ) -convex on I .(b) g ∈ D p ∩ K p + if and only if g ( r ) ∈ D p − r ∩ K p − r + .Proof. Assertion (a) follows from Lemma 2.2(c) and Lemma 2.2(g). To seethat assertion (b) holds, it is enough to show that, for any p > , we have g ∈ D p ∩ K p + if and only if g ′ ∈ D p − ∩ K p − + . Suppose first that g ∈ D p ∩ K p + .Then g ′ ∈ K p − + by assertion (a). Let x > be so that g is p -convex on [ x , ∞ ) .By the mean value theorem, there exist ξ x , ξ x ∈ (
0, 1 ) such that ∆ p g ( x − ) = ∆ p − g ′ ( x − + ξ x ) ∆ p − g ′ ( x ) ∆ p − g ′ ( x + ξ x ) = ∆ p g ( x ) . Letting x → ∞ , we see that g ′ ∈ D p − R . Conversely, suppose that g ′ ∈ D p − ∩ K p − + . Then g ∈ K p + by assertion (a). Let x > be so that g ′ is ( p − ) -convexon [ x , ∞ ) and let t ∈ ( x , x + ) . Then we have ∆ p − g ′ ( x ) ∆ p − g ′ ( t ) ∆ p − g ′ ( x + ) . Integrating on t ∈ ( x , x + ) , we obtain ∆ p − g ′ ( x ) ∆ p g ( x ) ∆ p − g ′ ( x + ) . Letting x → ∞ , we see that g ∈ D p R . Theorem 4.9.
Let p ∈ N and g ∈ C p . Then g ∈ D p ∩ K p + (resp. g ∈ D p ∩ K p − )if and only if g ( p ) eventually increases (resp. decreases) to zero.Remark . The function g ( x ) = x sin x vanishes at infinity but its deriva-tive does not. Theorem 4.9 shows that if g ∈ C q ∩ D p ∩ K q for some p , q ∈ N such that p q , then all the functions g ( p ) , g ( p + ) , . . . , g ( q ) vanish at infinity.Proposition 4.8 does not provide any information on the derivative g ′ when g lies in C ∩ D ∩ K . The following proposition deals with this issue. Proposition 4.11. If g ∈ C ∩ D ∩ K − is such that g ′ ∈ K , then g ′ ∈ C ∩ e D − N ∩ K + . roof. Let x > be so that g is decreasing and g ′ is monotone on I x = [ x − ∞ ) .By the mean value theorem, there exist ξ x , ξ x ∈ (
0, 1 ) such that ∆g ( x − ) = g ′ ( x − + ξ x ) and ∆g ( x ) = g ′ ( x + ξ x ) . Thus, we have ∆g ( x − ) g ′ ( x ) ∆g ( x ) or ∆g ( x − ) > g ′ ( x ) > ∆g ( x ) according to whether g ′ is increasing or decreasing. In both cases, we see that g ′ ( x ) approaches zero as x → ∞ . Since g ′ is nonpositive on I x , it must beincreasing on I x ; hence g ′ ∈ K + . For any integers m , n such that x m n ,we then have g ( n − ) − g ( m − ) = n − X k = m ∆g ( k − ) n − X k = m g ′ ( k ) Letting n → N ∞ , we can see that g ′ ∈ e D − N . Remark . The assumption that g ′ ∈ K in Proposition 4.11 cannot beignored. Indeed, one can show that the function g ( x ) = x ( x + sin x ) lies in C ∩ D ∩ K − whereas its derivative g ′ does not lie in K .Combining Lemma 2.2(b) with Theorem 3.4 and Proposition 4.2(b), we canvery easily obtain the following two corollaries, in which the symbols R and D can be used interchangeably. Corollary 4.13.
Let g ∈ K p + (resp. g ∈ K p − ) for some p ∈ N . Then g ∈ D p S if and only if there exists a solution f : R + → R to the equation ∆f = g thatlies in D p + ∩ K p − (resp. D p + ∩ K p + ). Corollary 4.14.
For any p ∈ N , we have D p ∩ K p + ⊂ K p − − and D p ∩ K p − ⊂ K p − + . More precisely, if g ∈ D p and is p -convex (resp. p -concave) on anunbounded interval of R + , then on this interval it is also ( p − ) -concave(resp. ( p − ) -convex). We end this section by providing a characterization of the set R p ∩ K p = D p ∩ K p in terms of interpolating polynomials. Proposition 4.15.
Let g ∈ K p for some p ∈ N . Then we have g ∈ R p S ifand only if for any x
0, . . . , x p > , g [ a + x
0, . . . , a + x p ] → as a → S ∞ . Thislatter condition means that g asymptotically coincides with its interpolatingpolynomial with any p nodes.Proof. (Necessity) Suppose for instance that g lies in K p + and let s ∈ S , s > ,be so that g is p -convex on I s = [ s , ∞ ) . By Lemma 2.3, the map ( z
0, . . . , z p ) ∈ I p + s g [ z
0, . . . , z p ] is increasing in each place. Since g ∈ R p S , this map is alsononpositive on I p + s ; indeed, for s z · · · z p a , a ∈ S , and x > , wehave g [ z
0, . . . , z p ] g [ a , a +
1, . . . , a + p − a + x ] , a → S ∞ by (10). Now, for any x
0, . . . , x p > and any a > s + p , we have g [ a − p , . . . , a − a + x p ] g [ a + x
0, . . . , a + x p ] where the left side increases to zero as a → S ∞ .(Sufficiency) This immediately follows from Proposition 4.1. log Γ -type functions In this section we define and investigate the map, denote it by Σ , that carriesany function g ∈ ∪ p > ( D p ∩ K p ) into the unique solution f to the equation ∆f = g that arises from the existence Theorem 3.4. We also investigate certainproperties of these solutions, that we call multiple log Γ -type functions. Σ We define the asymptotic degree of a function f ∈ K to be the integer value deg f = min { q ∈ N : f ∈ D q R } − For instance, if f is a polynomial of degree p for some p ∈ N , then deg f = p . If f ( x ) = or f ( x ) = ln ( + /x ) , then deg f = − . If f ( x ) = x + sin x or f ( x ) = x ,then deg f = ∞ . It is easy to see that the identity deg f = + deg ∆f holdswhenever deg f > . However, it is no longer true when deg f = − . Note alsothat deg f should not to be confused with the limiting value of x∆f ( x ) /f ( x ) as x → ∞ , which is related to the notion of elasticity of a function.Now, define the map Σ : [ p > ( D p ∩ K p ) → ran ( Σ ) by the condition g ∈ D p ∩ K p ⇒ Σg ( x ) = lim n → ∞ f pn [ g ]( x ) , (23)where ran ( Σ ) denotes the range of Σ .This map is well defined; indeed, if g ∈ ( D p ∩ K p ) ∩ ( D q ∩ K q ) for some p < q , then by Proposition 3.5 the sequences n f pn [ g ]( x ) and n f qn [ g ]( x ) have the same limit. Thus, in view of Proposition 4.4, we can see that condition(23) holds for p = + deg g .Just as the indefinite integral of a function g is the class of functions whosederivative is g , the indefinite sum of a function g is the class of functions whosedifference is g (see, e.g., [34, p. 48]). The map Σ now enables one to refine thelatter definition as follows. 25 efinition 5.1. The principal indefinite sum of a function g ∈ ∪ p > ( D p ∩ K p ) is the class of functions c + Σg , where c ∈ R .We will sometimes add a subscript to specify the variable on which the map Σ acts. For instance, Σ x g ( x ) stands for the function obtained by applying Σ to the function x g ( x ) while Σg ( x ) stands for the value of the function Σg at x .Let us now examine some immediate properties of the map Σ . Theorems 3.1and 3.4 and Proposition 4.6 show that, for any p ∈ N and any g ∈ D p ∩ K p ,the function Σg has the following features: (cid:15) it lies in D p + ∩ K p = R p + ∩ K p , and (cid:15) it is the unique solution to the equation ∆f = g that lies in K p andvanishes at .We also have that deg Σg = + deg g whenever deg g > ; but this propertyno longer holds if deg g = − . Finally, using (11) we immediately see that therestriction of Σg to N ∗ is Σg ( n ) = n − X j = g ( j ) , n ∈ N ∗ . (24)Now, it is clear that the map Σ is one-to-one and it is even a bijectionsince we have restricted its codomain to its range. We then have the followingimmediate result. Proposition 5.2.
The map Σ is a bijection and its inverse is the restrictionof the difference operator ∆ to ran ( Σ ) .Remark . Quite surprisingly, we observe that if g ∈ D p ∩ K p for some p ∈ N ,then Σg need not lie in K p + (and hence the converse of Lemma 2.2(b) doesnot hold). For instance, for any c ∈ R , the function f ( x ) = c + − x ( +
13 sin x ) lies in K − \ K . Indeed, x f ′ ( x ) is π -periodic and negative while x f ′′ ( x ) is π -periodic and change in sign from x = π to x = π . However, the function g = ∆f lies in D ∩ K + for x ∆f ′ ( x ) is π -periodic and positive. log Γ -type functions Barnes [11–13] introduced a sequence of functions Γ Γ
2, . . . , called multiplegamma functions , that generalize the Euler gamma function. The restrictionsof these functions to R + are characterized by the equations Γ p + ( x + ) = Γ p + ( x ) Γ p ( x ) , Γ ( x ) = Γ ( x ) , Γ p ( ) = x > p ∈ N ∗ , (− ) p + D p + Γ p ( x ) > x > For recent references, see, e.g., Adamchik [1, 2] and Srivastava and Choi [76].Thus defined, this sequence satisfies the conditions ln Γ p + = − Σ ln Γ p and deg ln Γ p = p . Also, it can be naturally extended to the case when p = by setting Γ ( x ) = /x .When g ∈ D p ∩ K p and deg g = p − for some p ∈ N , we say that exp ◦ Σg (resp. Σg ) is a Γ p -type function (resp. a log Γ p -type function ). When p > , exp ◦ Σg reduces to the function Γ p when exp ◦ g is precisely the function /Γ p − ,which simply shows that the function Γ p restricted to R + is itself a Γ p -typefunction. We also let Γ p (resp. Log Γ p ) denote the set of Γ p -type functions (resp. log Γ p -type functions). Thus, by definition we have ran ( (cid:6) ) = [ p > ( (cid:6) | D p ∩ K p ) = [ p > Γ p . Finally, we say that a function f : R + → R is a multiple Γ -type function (resp. multiple log Γ -type function ) if it lies in ∪ p > Γ p (resp. ∪ p > Γ p ).Thus defined, the set of log Γ p -type functions can be characterized as follows. Proposition 5.4.
For any function f : R + → R and any p ∈ N , the followingassertions are equivalent.(i) f ∈ Log(cid:0) p .(ii) f ( ) = , f ∈ K p , ∆f ∈ D p ∩ K p , and deg ∆f = p − .(iii) f = Σ∆f , ∆f ∈ D p ∩ K p , and deg ∆f = p − .(iv) f ∈ ran ( (cid:6) ) and deg ∆f = p − .(v) If p > , then f ∈ ran ( (cid:6) ) and deg f = p .If p = , then f ∈ ran ( (cid:6) ) and deg f ∈ { −
1, 0 } . It follows from Proposition 5.4 that a function f : R + → R lies in ran ( Σ ) ifand only if there exists p ∈ N such that f ( ) = , f ∈ K p , and ∆f ∈ D p ∩ K p .We also have the following result, which was proved by Webster [80, Theorem5.1] in the special case when p = . Proposition 5.5.
Let g g g ∈ D p ∩ K p for some p ∈ N , let a > , and let h : R + → R be defined by the equation h ( x ) = g ( x + a ) for x > . Then(a) g + g ∈ D p ∩ K p and Σ ( g + g ) = Σg + Σg ; b) if g − g ∈ D p ∩ K p , then Σ ( g − g ) = Σg − Σg ;(c) h ∈ D p ∩ K p and Σh ( x ) = Σ x g ( x + a ) = Σg ( x + a ) − Σg ( a + ) .Proof. Assertions (a) and (b) are immediate. To see that (c) holds, define afunction j : R + → R by the equation j ( x ) = Σg ( x + a ) − Σg ( a + ) for x > .Then j is a solution to the equation ∆j = h that lies in K p and satisfies j ( ) = .Hence, Σh = j , as required. Example 5.6 (see [80]) . Consider the function g : R + → R defined by g ( x ) = ln xx + a for some a > . Then we have g ∈ D ∩ K (and also g ∈ D ∩ K ) andProposition 5.5 shows that Σg ( x ) = ln Γ ( x ) Γ ( a + ) Γ ( x + a ) . Also, since g is concave on R + , we have that Σg is convex on R + . As Webster [80,p. 615] observed, this is “a not completely trivial result, but one immediate fromthe approach adopted here.” Example 5.7 (A rational function) . The function g ( x ) = x + x + x = x + x − ℜ (cid:18) x + i (cid:19) clearly lies in D ∩ K . Using Proposition 5.5, we then have Σg ( x ) = c + (cid:0) x (cid:1) + ψ ( x ) − ℜ ψ ( x + i ) for some c ∈ R , where ψ ( x + i ) = D x ln Γ ( x + i ) . Indeed, the function h ( x ) = ℜ (cid:18) x + i (cid:19) = xx + lies in D ∩ K while the function f ( x ) = ℜ ψ ( x + i ) lies in K and satisfies ∆f = h . log Γ -type functions The uniform convergence of the sequence n f pn [ g ] shows that the function Σg is continuous whenever so is g . More generally, we also have the followingresult. Proposition 5.8.
Let g ∈ C ∩ D p ∩ K p for some p ∈ N . Then the followingassertions hold.(a) Σg ∈ C ∩ D p + ∩ K p . b) Σg is integrable at if and only if so is g .(c) Let n ∈ N ∗ be so that g is p -convex or p -concave on [ n , ∞ ) . For any a x , the following inequality holds (cid:12)(cid:12)(cid:12)(cid:12) Z xa ( f pn [ g ]( t ) − Σg ( t )) dt (cid:12)(cid:12)(cid:12)(cid:12) Z xa ⌈ t ⌉ (cid:12)(cid:12)(cid:12)(cid:0) t − p (cid:1)(cid:12)(cid:12)(cid:12) dt | ∆ p g ( n ) | . Moreover, the following assertions hold.(c1) The sequence n R xa ( f pn [ g ]( t ) − Σg ( t )) dt converges to zero.(c2) The sequence n R xa ( f pn [ g ]( t ) + g ( t )) dt converges to Z xa ( Σg ( t ) + g ( t )) dt = Z xa Σg ( t + ) dt . (c3) For any m ∈ N ∗ , the sequence n R xa ( f pn [ g ]( t ) − f pm [ g ]( t )) dt converges to Z xa ( Σg ( t ) − f pm [ g ]( t )) dt . Proof.
Assertion (a) follows from the uniform convergence of the sequence n f pn [ g ] . Assertion (b) follows from assertion (a) and the identity Σg ( x + ) − Σg ( x ) = g ( x ) . Now, using (13) we see that the function Σg − f pn [ g ] = ρ p + n [ Σg ] is integrable at 0 and hence on ( a , x ) . The inequality of assertion (c) thenfollows from Theorem 3.4(b); and hence assertion (c1) also holds. Assertion(c2) follows from assertion (c1) and the identity Σg ( x + ) − Σg ( x ) = g ( x ) .Finally, using (18) we see that the function f pm [ g ] − f pn [ g ] is integrable on ( a , x ) and hence assertion (c3) follows from assertion (c1). In this section we essentially provide for multiple log Γ -type functions analoguesof Stirling’s formula , Stirling’s constant , and
Euler’s constant . We alsorevisit
Gregory’s summation formula and show how it can be derived almosttrivially in this context.
The asymptotic behavior of the gamma function for large values of its argumentcan be summarized as follows: for any a > , we have the following asymptoticequivalences Γ ( x + a ) ∼ x a Γ ( x ) as x → ∞ ; (25) Γ ( x ) ∼ √ π e − x x x − as x → ∞ . (26)29n this subsection we provide and discuss analogues of these formulas forthe multiple log Γ -type functions. We start with a technical but fundamentallemma. Recall first that, for any n ∈ N , the n th Gregory coefficient (also calledthe n th Bernoulli number of the second kind) is the number G n defined by theequation (see, e.g., [17–19, 61]) G n = Z (cid:0) tn (cid:1) dt . The first few values of G n are:
1, 12 , −
112 , 124 , − . These numbers aredecreasing in absolute value and satisfy the equations ∞ X n = | G n | = and G n = (− ) n − | G n | for n > (27)To simplify the notation, for any n ∈ N we set G n = − n X j = | G j | . By (27) we have G n > for all n ∈ N . Also, from the straightforward identity (− ) n (cid:0) t − n (cid:1) = − n X j = (− ) j − (cid:0) tj (cid:1) , we easily derive Z (cid:12)(cid:12)(cid:0) t − n (cid:1)(cid:12)(cid:12) dt = (− ) n Z (cid:0) t − n (cid:1) dt = (cid:12)(cid:12)(cid:12)(cid:12) Z (cid:0) t − n (cid:1) dt (cid:12)(cid:12)(cid:12)(cid:12) = G n . (28) Lemma 6.1.
Let g ∈ D p ∩ K p for some p ∈ N (hence g ∈ K p − if p > )and let a > .(a) Let x > be so that g is p -convex or p -concave on [ x , ∞ ) . Then | ρ p + x [ Σg ]( a ) | ⌈ a ⌉ (cid:12)(cid:12)(cid:12)(cid:0) a − p (cid:1)(cid:12)(cid:12)(cid:12) | ∆ p g ( x ) | , (29) and if g ∈ C , (cid:12)(cid:12)(cid:12)(cid:12) Z ρ p + x [ Σg ]( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) G p | ∆ p g ( x ) | . (30) (b) Suppose that p > and let x > be so that g is ( p − ) -convex or ( p − ) -concave on [ x , ∞ ) . Then | ρ px [ g ]( a ) | ⌈ a ⌉ (cid:12)(cid:12)(cid:12)(cid:0) a − p − (cid:1)(cid:12)(cid:12)(cid:12) | ∆ p g ( x ) | , (31) and if g ∈ C , (cid:12)(cid:12)(cid:12)(cid:12) Z ρ px [ g ]( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) G p − | ∆ p g ( x ) | . (32)30 roof. Assuming for instance that g is eventually p -convex, the function ∆ p g increases to zero on [ x , ∞ ) by Lemma 2.3. Using Lemma 2.4, we then obtain | ρ p + x [ Σg ]( a ) | (cid:12)(cid:12)(cid:12)(cid:0) a − p (cid:1)(cid:12)(cid:12)(cid:12) ⌈ a ⌉ − X j = | ∆ p g ( x + j ) | from which we immediately derive (29). Now, observing that the function t ρ p + x [ Σg ]( t ) does not change in sign on (
0, 1 ) by Lemma 2.4, and thenintegrating both sides of (29) on a ∈ (
0, 1 ) , we obtain (cid:12)(cid:12)(cid:12)(cid:12) Z ρ p + x [ Σg ]( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) = Z | ρ p + x [ Σg ]( t ) | dt Z (cid:12)(cid:12)(cid:12)(cid:0) t − p (cid:1)(cid:12)(cid:12)(cid:12) dt | ∆ p g ( x ) | , which, using (28), gives (30). We prove (31) and (32) similarly. A first asymptotic result. If g ∈ D p ∩ K p for some p ∈ N , then for any a ∈ N the a th degree Newton expansion of Σg ( x + a ) is given by Σg ( x + a ) = a X j = (cid:0) aj (cid:1) ∆ j Σg ( x ) , or equivalently, Σg ( x + a ) − Σg ( x ) − X j > (cid:0) aj (cid:1) ∆ j − g ( x ) = If the index variable j in the latter sum is bounded above by p , then clearlythe resulting left-hand expression need no longer be zero but it approaches zeroas x → ∞ (because g ∈ D p R ). The following theorem shows that this latterproperty still holds when a is any nonnegative real number, thus providing theasymptotic behavior of the difference Σg ( x + a ) − Σg ( x ) for large values of x .We omit the proof of this theorem for it follows immediately from (29) and(31). We also observe that the first convergence result (33) was established byWebster [80, Theorem 6.1] in the case when p = . The second one (34) simplyexpresses the fact that g lies in R p R by Proposition 4.5. Theorem 6.2.
Let g ∈ D p ∩ K p for some p ∈ N and let a > . Let also x > be so that g is p -convex or p -concave on [ x , ∞ ) . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Σg ( x + a ) − Σg ( x ) − p X j = (cid:0) aj (cid:1) ∆ j − g ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ⌈ a ⌉ (cid:12)(cid:12)(cid:12)(cid:0) a − p (cid:1)(cid:12)(cid:12)(cid:12) | ∆ p g ( x ) | , with equality if a ∈ {
1, 2, . . . , p } . In particular, Σg ( x + a ) − Σg ( x ) − p X j = (cid:0) aj (cid:1) ∆ j − g ( x ) → as x → ∞ . (33)31 f p > and if x > is so that g is ( p − ) -convex or ( p − ) -concave on [ x , ∞ ) , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g ( x + a ) − p − X j = (cid:0) aj (cid:1) ∆ j g ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ⌈ a ⌉ (cid:12)(cid:12)(cid:12)(cid:0) a − p − (cid:1)(cid:12)(cid:12)(cid:12) | ∆ p g ( x ) | , with equality if a ∈ {
1, 2, . . . , p − } . In particular, g ( x + a ) − p − X j = (cid:0) aj (cid:1) ∆ j g ( x ) → as x → ∞ . (34) Example 6.3.
Let us apply Theorem 6.2 to the function g ( x ) = ln x . For thisfunction we have p = + deg g = and Σg ( x ) = ln Γ ( x ) . Thus, for any x > and any a > we obtain (cid:18) + x (cid:19) − ⌈ a ⌉ | a − | Γ ( x + a ) Γ ( x ) x a (cid:18) + x (cid:19) ⌈ a ⌉ | a − | , (35)with equalities if a = . Thus, we retrieve the asymptotic equivalence (25).Interestingly, Wendel [81] provided the following tighter inequalities (cid:16) + ax (cid:17) a − Γ ( x + a ) Γ ( x ) x a if a Considering higher values of p may provide inequalities that are tighter than(35). For instance, taking p = , we obtain (cid:18) + x (cid:19) ( a ) − ⌈ a ⌉ | ( a − ) | (cid:18) + x (cid:19) ⌈ a ⌉ | ( a − ) | Γ ( x + a ) Γ ( x ) x a (cid:18) + x (cid:19) ( a ) + ⌈ a ⌉ | ( a − ) | (cid:18) + x (cid:19) − ⌈ a ⌉ | ( a − ) | . Thus, we can see that the central function in the inequalities above can alwaysbe “sandwiched” by finite products of powers of rational functions. Similarinequalities for this function can be found, e.g., in [76, pp. 106–107].
The asymptotic constant and Binet-like function.
With any function g lyingin ∪ p > ( C ∩ D p ∩ K p ) we associate the number σ [ g ] = Z Σg ( t + ) dt = Z ( Σg ( t ) + g ( t )) dt . (36)We then observe that the following identity holds for any x > , Z x + x Σg ( t ) dt = σ [ g ] + Z x g ( t ) dt . (37)32ndeed, both sides have the same derivative and the same value at x = .It would be convenient to name the constant σ [ g ] for we will make an in-tensive use of it throughout the rest of this paper. In view of Theorem 6.5below, we will call it the asymptotic constant associated with the function g ,although a more appropriate name for this constant could also be proposed andused in subsequent papers.Just as in Lemma 6.1, we have assumed the continuity of function g to ensurethat the integrals in (36) and (37) be defined. Of course, this assumption couldhave been relaxed into weaker properties such as local integrability of both g and Σg . However, for the sake of simplicity we will henceforth assume the continuityof any function whenever we need to integrate it on a compact interval; see alsoRemark 11.1.We also have the following proposition, which follows immediately fromProposition 5.5 and identity (37). Proposition 6.4.
Let g ∈ ∪ p > ( C ∩ D p ∩ K p ) , let a > , and let h : R + → R be defined by the equation h ( x ) = g ( x + a ) for x > . Then σ [ h ] = σ [ g ] + Z + a g ( t ) dt − Σg ( a + ) . Now, for any q ∈ N and any g ∈ C , we define the function J q [ g ] : R + → R by the equations J q [ g ]( x ) = − Z ρ qx [ g ]( t ) dt = q − X j = G j ∆ j g ( x ) − Z x + x g ( t ) dt . (38)When g ( x ) = ln Γ ( x ) and q = , this function reduces to Binet’s function J ( x ) related to ln Γ ( x ) (see, e.g., [28, p. 224]). That is, J [ ln Γ ]( x ) = J ( x ) = ln Γ ( x ) −
12 ln ( π ) + x − (cid:18) x − (cid:19) ln x . We will say that the function J q [ g ] is the Binet-like function associated withthe function g and the parameter q . As we will see in the rest of this paper,many subsequent definitions and results will be expressed in terms of the Binet-like function.Using (37), we can also see that for any q ∈ N and any g ∈ ∪ p > ( C ∩ D p ∩ K p ) we have J q + [ Σg ]( x ) = Σg ( x ) − Z x g ( t ) dt + q X j = G j ∆ j − g ( x ) − σ [ g ] . (39)In particular, we have σ [ g ] = − J [ Σg ]( ) and ∆J q + [ Σg ] = J q + [ g ] . eneralized Stirling’s formula. The following important theorem enables oneto investigate, for any function g lying in ∪ p > ( C ∩ D p ∩ K p ) , the asymptoticbehavior of the function Σg for large values of its argument. In particular, theconvergence result (40) gives for Σg an analogue of Stirling’s formula. We call itthe generalized Stirling formula . Combining (33) with (40) then immediatelyprovides the asymptotic behavior of Σg ( x + a ) for any a > . We also observethat alternative formulations of (40) in the case when p = were establishedby Krull [46, p. 368] and later by Webster [80, Theorem 6.3]. Theorem 6.5 (Generalized Stirling’s formula) . Let g ∈ C ∩ D p ∩ K p for some p ∈ N and let x > be so that g is p -convex or p -concave on [ x , ∞ ) . Then (cid:12)(cid:12) J p + [ Σg ]( x ) (cid:12)(cid:12) G p | ∆ p g ( x ) | . In particular, the function J p + [ Σg ] lies in D R , that is, Σg ( x ) − Z x g ( t ) dt + p X j = G j ∆ j − g ( x ) → σ [ g ] as x → ∞ . (40) If p > and if x > is so that g is ( p − ) -convex or ( p − ) -concave on [ x , ∞ ) , then | J p [ g ]( x ) | G p − | ∆ p g ( x ) | . In particular, the function J p [ g ] lies in D R , that is, − Z x + x g ( t ) dt + p − X j = G j ∆ j g ( x ) → as x → ∞ . (41) Proof.
The first inequality is obtained from (30), (37), and (39). The firstconvergence result (40) immediately follows. The second inequality and itsassociated convergence result (41) is obtained similarly using (32) and (38).
Remark . We can readily see that (34) can be obtained by applying theoperator ∆ x to (33). More generally, the first part of Theorem 6.2 can beobtained by replacing g by Σg and p by p + in the second part. The sameobservation applies to Theorem 6.5 (cf. the identity ∆J p + [ Σg ] = J p + [ g ] ). Example 6.7.
Applying Theorem 6.5 to the function g ( x ) = ln x with p = ,we immediately obtain the following inequalities for any x > (cid:18) + x (cid:19) − Γ ( x ) √ π e − x x x − (cid:18) + x (cid:19)
12 ; (cid:18) + x (cid:19) x e (cid:18) + x (cid:19) x + Γ ( x ) ∼ √ π e − x x x − as x → ∞ ; x ! = Γ ( x + ) ∼ √ πx e − x x x as x → ∞ ; (cid:0) + x (cid:1) x ∼ e as x → ∞ . Just as in Example 6.3, tighter inequalities can be obtained by consideringhigher values of p . For instance, for p = , we obtain (cid:18) + x (cid:19) − (cid:18) + x (cid:19) Γ ( x ) √ π e − x x x − (cid:18) + x (cid:19) (cid:18) + x (cid:19) −
512 .
For p = , we obtain (cid:18) + x (cid:19) − (cid:18) + x (cid:19) (cid:18) + x (cid:19) − Γ ( x ) √ π e − x x x − (cid:18) + x (cid:19) (cid:18) + x (cid:19) − (cid:18) + x (cid:19)
38 .
Thus, we see that the central function in these inequalities can always be brack-eted by finite products of radical functions.Example 6.7 illustrates the possibility of obtaining closer bounds for theBinet-like function J p + [ Σg ]( x ) by considering any value of p that is higher than + deg g . Actually, it is not difficult to see that this feature applies to everycontinuous multiple log Γ -type function. We discuss this issue in Appendix Cand show that the inequalities actually get tighter and tighter as p increases. Improvements of Stirling’s formula.
The following estimate of the gammafunction is due to Gosper [33] Γ ( x ) ∼ √ π e − x x x − (cid:18) + x (cid:19) as x → ∞ , and is more accurate than Stirling’s formula. On the basis of this alternativeapproximation, Mortici [64] provided the following narrow inequalities (cid:16) + α x (cid:17) < Γ ( x ) √ π e − x x x − < (cid:18) + β x (cid:19)
12 , for x > where α = ≈ and β = ( / ) / − ≈ . We actually observethat the quest for finer and finer bounds and approximations for the gammafunction has gained an increasing interest during this last decade (see [23,25,26,31, 56, 63–66, 82, 83] and the references therein). We believe that some of theseinvestigations could be generalized to various Γ p -type functions. New resultsalong this line would be most welcome.35 eries expressions for Σg and σ [ g ] . The following result provides series ex-pressions for Σg ( x ) and σ [ g ] in terms of Gregory’s coefficients (see also Propo-sition C.2 in Appendix C). Proposition 6.8.
Let g ∈ C ∩ D p ∩ K ∞ for some p ∈ N . Let x > be sothat for every integer q > p the function g is q -convex or q -concave on [ x , ∞ ) . Suppose also that the sequence q ∆ q g ( x ) is bounded. Then thesequence n G n ∆ n − g ( x ) is summable and we have J ∞ [ Σg ]( x ) = , i.e., Σg ( x ) = σ [ g ] + Z x g ( t ) dt − ∞ X n = G n ∆ n − g ( x ) . In particular, if the assumptions above are satisfied for x = , then we have σ [ g ] = ∞ X n = G n ∆ n − g ( ) . (42) Proof.
Since the sequence n G n converges to zero, by (30) so does thesequence q Z ρ q + x [ Σg ]( t ) dt . We then obtain the result using (37).
Example 6.9.
Applying Proposition 6.8 to the function g ( x ) = ln ( x ) , we obtainthe following infinite product representation of the gamma function Γ ( x ) = √ π e − x x x − ∞ Y n = e − G n ∆ n − ( x ) , that is, Γ ( x ) = √ π e − x x x − (cid:18) x + x (cid:19) (cid:18) ( x + ) x ( x + ) (cid:19) − × (cid:18) ( x + )( x + ) ( x + ) x (cid:19) A similar representation of the gamma function can be found in Feng andWang [31].
Fontana-Mascheroni’s series.
When g ( x ) = x and p = , identity (42) re-duces to the well-known formula γ = ∞ X n = | G n | n , Fontana-Mascheroni’s series (see, e.g., [17]).Thus, the series representation of the asymptotic constant σ [ g ] given in (42)provides the analogue of Fontana-Mascheroni’s series for any function g satis-fying the assumptions of Proposition 6.8.The following proposition provides a way to construct a function g ( x ) thathas a prescribed asymptotic constant σ [ g ] given in the form (42). Proposition 6.10.
Let S = P ∞ n = G n s n for some sequence n s n and let g : R + → R be such that g ( n ) = n X k = (cid:0) n − k − (cid:1) s k , n ∈ N ∗ . If g satisfies the assumptions of Proposition 6.8, then the following asser-tions hold.(a) S = σ [ g ] .(b) Σg ( n ) = P n − k = (cid:0) n − k (cid:1) s k for any n ∈ N ∗ .(c) s n = ∆ n − g ( ) = ∆ n Σg ( ) for any n ∈ N ∗ .Proof. Using the classical inversion formula [34, p. 192], we obtain s n + = n X k = (− ) n − k (cid:0) nk (cid:1) g ( k + ) = ∆ n g ( ) . This establishes assertion (c) and then assertion (a) by Proposition 6.8. Asser-tion (b) is straightforward using (24).
Example 6.11.
Let us apply Proposition 6.10 to the series S = ∞ X n = | G n | n Let g : R + → R be a function such that g ( n ) = n X k = (− ) k − (cid:0) n − k − (cid:1) k = n H n , n ∈ N ∗ . We naturally take g ( x ) = x H x , from which we derive Σg ( x ) = π − ψ ( x ) + H x − Thus, we have S = σ [ g ] . Combining this with the definition of σ [ g ] , we derivethe surprising identity Z H t dt = − π + ∞ X n = | G n | n Z H t dt = γ = ∞ X n = | G n | n . To give another example, consider the series S = ∞ X n = | G n | n + a , where a > . Proposition 6.10 shows that we can take g ( x ) = B ( x , a + ) and Σg ( x ) = a − B ( x , a ) , where ( x , y ) B ( x , y ) is the beta function. We then derive the identity ∞ X n = | G n | n + a = a − Z
10 B ( x + a ) dx . Using the definition of the beta function as an integral, this identity also reads ∞ X n = | G n | n + a = a + Z x a ln ( − x ) dx . Setting a = for instance, we obtain ∞ X n = | G n | n + = + Z √ x ln ( − x ) dx and the decimal expansion of the latter integral is Sloane’s A094691 [75]. Asymptotic behaviors and trends.
The following corollary, which immediatelyfollows from (37) and (40), particularizes the generalized Stirling formula whenthe function g lies in C ∩ D ∩ K . Corollary 6.12.
Let g ∈ C ∩ D ∩ K . Then Σg ( x ) − Z x + x Σg ( t ) dt → as x → ∞ . Equivalently, Σg ( x ) − Z x g ( t ) dt → σ [ g ] as x → ∞ .
38t is clear that the integral (37) planes and cancels out the cyclic variationsof any -periodic additive component of Σg in the sense that the function x Z x + x ω ( t ) dt is constant for any -periodic function ω : R + → R . In fact, the integral (37) canbe interpreted as the trend of the function Σg , just as a moving average enablesone to decompose a time series into its trend and its seasonal variation. Thus,Corollary 6.12 shows that Σg ( x ) coincides asymptotically with its trend (as wecould expect from a function lying in C ∩ D ∩ K ) and behaves asymptoticallylike the antiderivative of g .The centered version of integral (37), namely Z x + x − Σg ( t ) dt = σ [ g ] + Z x − g ( t ) dt , x >
12 , naturally provides a more accurate trend of Σg . The following corollary showsthat Σg ( x ) coincides asymptotically with this latter trend whenever g lies in C ∩ D ∩ K or in C ∩ D ∩ K . It is not difficult to see that in general thisresult no longer holds when g lies in C ∩ D ∩ K . The logarithm of the Barnes G -function (see Subsection 9.5) could serve as an example here. Corollary 6.13.
Let p ∈ {
0, 1 } , let g ∈ C ∩ D p ∩ K p , and let x > be sothat g is p -convex or p -concave on [ x , ∞ ) . Then (cid:12)(cid:12)(cid:12)(cid:12) Σg (cid:18) x + (cid:19) − Z x + x Σg ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12) J p + [ Σg ]( x ) (cid:12)(cid:12) G p | ∆ p g ( x ) | . In particular, Σg ( x ) − Z x + x − Σg ( t ) dt → as x → ∞ , or equivalently, Σg ( x ) − Z x − g ( t ) dt → σ [ g ] as x → ∞ . Proof.
Using Theorem 6.5, we see that it is enough to prove the first inequality.Let h ( x ) = Σg (cid:18) x + (cid:19) − Z x + x Σg ( t ) dt . Consider first the case when p = and suppose for instance that g lies in K + ;39ence Σg is decreasing on [ x , ∞ ) . Then it is geometrically clear that | h ( x ) | Z x + x Σg ( t ) dt − Σg (cid:18) x + (cid:19) , if h ( x ) Σg (cid:18) x + (cid:19) − Z x + x + Σg ( t ) dt , if h ( x ) > and that both quantities are less than or equal to J [ Σg ]( x ) .Now, suppose that p = and for instance that g lies in K + ; hence Σg is concave on [ x , ∞ ) . Using the Hermite-Hadamard inequality and then thetrapezoidal rule on the intervals [ x , x + ] and [ x +
12 , x + ] , we obtain thefollowing chain of inequalities: h ( x ) Z x + x Σg ( t ) dt − Σg ( x + ) − Σg ( x ) and the latter quantity is exactly − J [ Σg ]( x ) .Applying Corollary 6.13 to the function g ( x ) = ln x , we obtain Burnside’sformula [24] (see also [63]) Γ ( x ) ∼ √ π (cid:18) x − e (cid:19) x − as x → ∞ . (43)Thus, Corollary 6.13 gives an analogue of Burnside’s formula for any continuous Γ p -type function when p ∈ {
0, 1 } . It also shows that this formula provides abetter approximation than the generalized Stirling formula for all functions g lying in C ∩ D p ∩ K p with p ∈ {
0, 1 } . The number √ π arising in Example 6.7 is called Stirling’s constant (see, e.g.,[32]). For certain multiple Γ -type functions, analogues of Stirling’s constant canbe easily defined as follows. Definition 6.14 (Generalized Stirling’s constant) . For any function g lying in ∪ p > ( C ∩ D p ∩ K p ) and integrable at , we define the number σ [ g ] = σ [ g ] − Z g ( t ) dt = Z Σg ( t ) dt . We say that the number exp ( σ [ g ]) is the generalized Stirling constant associ-ated with g . 40ote that, contrary to the generalized Stirling constant, the asymptoticconstant σ [ g ] exists for any function g lying in ∪ p > ( C ∩ D p ∩ K p ) , even if itis not integrable at . This shows that the asymptotic constant is the “good”constant to consider in this new theory. When g is integrable at , then (40)can take the form Σg ( x ) − Z x g ( t ) dt + p X j = G j ∆ j − g ( x ) → σ [ g ] as x → ∞ . Let g ∈ C , q ∈ N , and let m n be integers. Integrating both sides of(18) on x ∈ (
0, 1 ) , we obtain Z nm g ( t ) dt = n − X k = m g ( k ) + q X j = G j ( ∆ j − g ( n ) − ∆ j − g ( m )) + R q , m , n , (44)where R q , m , n = Z n − X k = m ρ q + k [ g ]( t ) dt = Z ( f qm [ g ]( t ) − f qn [ g ]( t )) dt . Identity (44) is nothing other than
Gregory’s summation formula (see,e.g., [14, 43, 62]) with an integral form of the remainder. Note that, just likeidentity (8), equation (44) is a pure identity and therefore holds without anyrestriction on the form of g ( x ) . Actually, this identity can be simply written interms of the Binet-like function as n − X k = m J q + [ g ]( k ) = − R q , m , n , or equivalently, if g ∈ ∪ p > ( C ∩ D p ∩ K p ) , J q + [ Σg ]( n ) − J q + [ Σg ]( m ) = − R q , m , n . Remark . We observe that Jordan [43, p. 285] established that“ R q , m , n = G q + ( n − m ) ∆ q + g ( ξ ) ”for some ξ ∈ ( m , n ) . However, taking for instance g ( x ) = x and ( q , m , n ) =(
0, 1, 2 ) shows that this form of the remainder is not correct. However, weconjecture that Jordan’s formula can be corrected by assuming that ξ ∈ ( m − n − ) . 41he following lemma, which is an immediate consequence of Lemma 2.4,provides an upper bound for | R p , m , n | when g is p -convex or p -concave on [ m , ∞ ) .We then see that, under this latter assumption, Gregory’s formula (44) providesa quadrature formula for the numerical computation of the integral of g overthe interval ( m , n ) . Lemma 6.16.
Let g ∈ C ∩ K p for some p ∈ N and let m ∈ N ∗ be so that g is p -convex or p -concave on [ m , ∞ ) . Then, for any integer n > m , wehave | R p , m , n | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − X k = m Z ρ p + k [ g ]( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G p | ∆ p g ( n ) − ∆ p g ( m ) | . (45) Example 6.17.
Let us compute the integral I = Z ππ ln x dx = numerically using Gregory’s summation formula (44). Using an appropriatelinear change of variable, we obtain I = R n g ( t ) dt , where g ( t ) = πn − (cid:0) πn − ( t − ) + π (cid:1) . Taking n = and q = for instance, we obtain I ≈ X k = g ( k ) + X j = G j ( ∆ j − g ( ) − ∆ j − g ( )) = and (45) gives | R | × − .In the following result, we give sufficient conditions on function g for thesequence q R q , m , n to converge to zero. Gregory’s formula (44) then takes aspecial form. Proposition 6.18.
Let g ∈ C , p ∈ N , and let m n be integers.Suppose that, for every integer q > p , the function g is q -convex or q -concave on [ m , ∞ ) . Suppose also that the sequence q ∆ q g ( n ) − ∆ q g ( m ) is bounded. Then we have Z nm g ( t ) dt = n − X k = m g ( k ) + ∞ X j = G j ( ∆ j − g ( n ) − ∆ j − g ( m )) , or equivalently, n − X k = m J ∞ [ g ]( k ) = If g ∈ ∪ p > ( C ∩ D p ∩ K p ) , then this latter condition simply means that J ∞ [ Σg ]( n ) = J ∞ [ Σg ]( m ) . roof. The sequence q R q , m , n converges to zero by (45). The result thenimmediately follows from Gregory’s formula (44). Example 6.19.
Taking g ( x ) = ln x and m = p = in Proposition 6.18, weobtain the following identity for any n ∈ N ∗ ln n ! = − n + (cid:18) n + (cid:19) ln n +
112 ln (cid:18) n + n (cid:19) −
124 ln (cid:18) n ( n + ) ( n + ) (cid:19) + · · · Gregory’s formula (44) is sometimes presented in a more general form in theliterature. We provide this general form in the following proposition using ourintegral expression for the remainder. Lemma 6.16 then can be easily adaptedto this general form.
Proposition 6.20 (General form of Gregory’s formula) . Let a ∈ R , n , q ∈ N , h > , and f ∈ C ([ a , ∞ )) . Then h Z a + nha f ( t ) dt = n − X k = f ( a + kh )+ q X j = G j (cid:16) ( ∆ j − [ h ] f )( a + nh ) − ( ∆ j − [ h ] f )( a ) (cid:17) + R hq , a , n , where R hq , a , n = Z n X k = ρ p + k [ g ]( t ) dt and g ( x ) = f ( a + ( x − ) h ) . Here, ∆ [ h ] denotes the forward difference operator with step h > .Proof. This formula can be obtained immediately from (44) replacing n by n + and then setting m = and g ( x ) = f ( a + ( x − ) h ) .Gregory’s formula is often compared with the corresponding Euler-Maclau-rin summation formula . We now recall the latter in its general form (see,e.g., [76, p. 220]) as we will use it a few times in this paper. We also notethat Euler-Maclaurin’s formula is more advantageous than Gregory’s formula ifwe deal with functions whose derivatives are less complicated than their differ-ences. However, there are functions for which Euler-Maclaurin’s formula leadsto divergent series while the corresponding Gregory’s formula-based series (seeProposition 6.18) are convergent. For instance, this may be due to the fact that D n x increases indefinitely with n while ∆ n x tends to zero if n increases. (Here,we paraphrase from Jordan [43, p. 285].)43 roposition 6.21 (Euler-Maclaurin formula) . Let N ∈ N ∗ , f ∈ C ([ a , b ]) , and h = ( b − a ) /N , for some real numbers a < b . Then we have h N X k = f ( a + kh ) = Z ba f ( x ) dx + h ( f ( a ) + f ( b ))+ h Z N B ( s − ⌊ s ⌋ ) Df ( a + sh ) ds . If, in addition, f ∈ C q ([ a , b ]) for some q ∈ N ∗ , then h N X k = f ( a + kh ) = Z ba f ( x ) dx + h ( f ( a ) + f ( b ))+ q X j = h j B j ( j ) ! (cid:0) D j − f ( b ) − D j − f ( a ) (cid:1) + R , where R = − h q + Z N B q ( s − ⌊ s ⌋ )( q ) ! D q f ( a + sh ) ds and | R | h q | B q | ( q ) ! Z ba | D q f ( x ) | dx . Here f ∈ C k ([ a , b ]) means that f ∈ C k ( I ) for some open interval I containing [ a , b ] . Suppose that g ∈ C ∩ D p ∩ K p for some p ∈ N . Let also m ∈ N ∗ be so that g is p -convex or p -concave on [ m , ∞ ) . By Lemma 6.16 the sequence n R p , m , n converges and we have (see also Proposition 5.8) R p , m , ∞ = lim n → ∞ Z ( f pm [ g ]( t ) − f pn [ g ]( t )) dt = ∞ X k = m Z ρ p + k [ g ]( t ) dt = Z ∞ X k = m ρ p + k [ g ]( t ) dt = Z ( f pm [ g ]( t ) − Σg ( t )) dt = − Z ρ p + m [ Σg ]( t ) dt = J p + [ Σg ]( m ) , where the fifth equality follows from (13). Also, (45) reduces to | R p , m , ∞ | = | J p + [ Σg ]( m ) | G p | ∆ p g ( m ) | , (46)which is also an immediate consequence of Theorem 6.5.44et us now provide a geometric interpretation of the remainder R p , m , ∞ .Suppose for instance that g is p -concave on [ m , ∞ ) and that p is even; theother cases are similar. Then by (9) and Lemma 2.4, for any integer k ∈ [ m , ∞ ) and any t ∈ (
0, 1 ) , we have > ρ p + k [ g ]( t ) = g ( k + t ) − P p [ g ]( k , k +
1, . . . , k + p ; k + t ) , which means that, on the interval [ k , k + ] , the graph of g lies under (or on)that of its interpolating polynomial with nodes at k , k +
1, . . . , k + p . Also, the(signed) surface area between both graphs is − Z ρ p + k [ g ]( t ) dt = J p + [ g ]( k ) . Summing this area for k = m , . . . , n − and letting n → N ∞ , we obtain thecumulated (signed) surface area − ∞ X k = m Z ρ p + k [ g ]( t ) dt = ∞ X k = m J p + [ g ]( k ) = − R p , m , ∞ . (47)This interpretation is particularly visual when p = or p = . For instance,when p = , the graph of g on [ m , ∞ ) lies either over or under the polygonal linethrough the points ( k , g ( k )) for all integers k > m . The value − R p , m , ∞ is thenthe signed area between the graph of g and this polygonal line and correspondsto the remainder in the trapezoidal rule on [ m , ∞ ) .In view of this interpretation, we now propose the following definition. Definition 6.22 (Generalized Euler’s constant) . The generalized Euler con-stant associated with a function g ∈ ∪ p > ( C ∩ D p ∩ K p ) is the number γ [ g ] = − R p ,1, ∞ = ∞ X k = J p + [ g ]( k ) = − J p + [ Σg ]( ) , where p = + deg g .For instance, if g ∈ C ∩ D ∩ K , we have γ [ g ] = lim n → ∞ n − X k = g ( k ) − Z n g ( t ) dt ! (48) = ∞ X k = (cid:18) g ( k ) − Z k + k g ( t ) dt (cid:19) , which represents the remainder in the rectangle method on [ ∞ ) .45imilarly, if g ∈ C ∩ D ∩ K and deg g = , we get γ [ g ] = lim n → ∞ n − X k = g ( k ) − Z n g ( t ) dt + g ( n ) − g ( ) ! = ∞ X k = (cid:18) g ( k ) − Z k + k g ( t ) dt + ∆g ( k ) (cid:19) , which represents the remainder in the trapezoidal rule on [ ∞ ) . If g ∈ C ,then this latter expression also reduces to (use integration by parts) γ [ g ] = ∞ X k = Z k + k (cid:18) t − k − (cid:19) g ′ ( t ) dt = Z ∞ (cid:18) { t } − (cid:19) g ′ ( t ) dt , (49)where { t } = t − ⌊ t ⌋ .Thus defined, the number γ [ g ] generalizes to any function lying in ∪ p > ( C ∩ D p ∩ K p ) not only the classical Euler constant γ but also the generalized Eulerconstant γ [ g ] associated with a positive and strictly decreasing function g (see,e.g., [7, 32]), as defined in (48). Moreover, as we will see in Subsection 8.2, thisnumber plays a central role in the Weierstrassian form of Σg (which also justifiesthe choice m = in the definition of γ [ g ] ).If g is p -convex or p -concave on [ ∞ ) , then by (46) we also have the in-equality | γ [ g ] | G p | ∆ p g ( ) | . (50) A conversion formula between γ [ g ] and σ [ g ] . The following proposition,which immediately follows from the identity γ [ g ] = − J p + [ Σg ]( ) , shows howthe numbers γ [ g ] and σ [ g ] are related and provides an alternative way to com-pute the value of γ [ g ] . Proposition 6.23.
For any function g lying in ∪ p > ( C ∩ D p ∩ K p ) , we have σ [ g ] = γ [ g ] + p X j = G j ∆ j − g ( ) , where p = + deg g .An analogue of Liu’s formula for Γ -type functions. Using (49) together withProposition 6.21 with a = h = and b = N = n (first-order version of theEuler-Maclaurin formula), we obtain the following statement. For any n ∈ N ∗ and any g ∈ C ∩ D ∩ K , with deg g = , we have n X k = g ( k ) = γ [ g ] + Z n g ( t ) dt + ( g ( ) + g ( n ))+ Z ∞ n (cid:18) − { t } (cid:19) g ′ ( t ) dt , n X k = g ( k ) = σ [ g ] + Z n g ( t ) dt + g ( n ) + Z ∞ n (cid:18) − { t } (cid:19) g ′ ( t ) dt , (51)or equivalently, J [ Σg ]( n ) = Z ∞ n (cid:18) − { t } (cid:19) g ′ ( t ) dt , where { t } = t − ⌊ t ⌋ . Applying this result to g ( n ) = ln n , we obtain Liu’s exactformula [55] (see also [63]) n ! = √ πn (cid:16) ne (cid:17) n exp (cid:18) Z ∞ n − { t } t dt (cid:19) , n ∈ N ∗ . A generalization of (51) to any function g lying in ∪ p > ( C ∩ D p ∩ K p ) wouldbe welcome. An integral form of γ [ g ] . The following proposition shows that the integralrepresentation of the Euler constant γ = Z ∞ (cid:18) ⌊ t ⌋ − t (cid:19) dt can be generalized to the constant γ [ g ] for any function g lying in ∪ p > ( C ∩ D p ∩ K p ) . This result is a straightforward consequence of (47). Proposition 6.24.
For any g ∈ C ∩ D p ∩ K p , where p = + deg g , we have γ [ g ] = Z ∞ (cid:18) p X j = G j ∆ j g ( ⌊ t ⌋ ) − g ( t ) (cid:19) dt . In particular, when deg g = − , we have γ [ g ] = R ∞ ( g ( ⌊ t ⌋ ) − g ( t )) dt . We end this section by establishing two additional asymptotic results. Thefirst one concerns only the case when the sequence n g ( n ) is summable.The second one is much more general and concerns all the continuous multiple log Γ -type functions. We also discuss the search for simple conditions on function g : R + → R to ensure the existence of Σg . The case when g ( n ) is summable. In the special case when g ∈ e D − N ∩ K , thegeneralized Stirling formula and the constants γ [ g ] and σ [ g ] take very specialforms. We present them in the following proposition, which immediately followsfrom Theorem 3.7, Eq. (48), and Proposition 6.23.47 roposition 6.25. If g ∈ e D − N ∩ K , then we have Σg ( x ) → ∞ X k = g ( k ) as x → ∞ . If, in addition we have g ∈ C , then g is integrable at infinity and σ [ g ] = γ [ g ] = ∞ X k = g ( k ) − Z ∞ g ( t ) dt . A general asymptotic result.
The following general result gives a sufficientcondition on a function g lying in ∪ p > ( C ∩ D p ∩ K p ) for Σg to be asymptoticallyequivalent to its (possibly shifted) trend. Proposition 6.26.
Let g ∈ C ∩ D p ∩ K p for some p ∈ N and let a > . If p > , we assume that the function x R x + x Σg ( t ) dtΣg ( x + a ) (52) (that is defined in a neighborhood of infinity) is eventually monotone. Thenwe have Σg ( x + a ) ∼ Z x + x Σg ( t ) dt as x → ∞ . (53) Proof.
Assume first that g lies in C ∩ D ∩ K . Let us prove that for any c ∈ R we have c + R x + x Σg ( t ) dtc + Σg ( x + a ) → as x → ∞ . (54)Suppose that g lies in e D − N and for instance that Σg is eventually increasing.Then for sufficiently large x we have c + Σg ( x ) | c + Σg ( x + a ) | c + R x + x Σg ( t ) dt | c + Σg ( x + a ) | c + Σg ( x + ) | c + Σg ( x + a ) | and (54) then follows from Corollary 6.25. Suppose now that g does not liein e D − N , which implies that g is not integrable at infinity. It follows that theintegral (37) tends to infinity as x → ∞ , and hence so does the function Σg ( x ) .In this case, by (33) and (40) we obtain Σg ( x + a ) − Z x + x Σg ( t ) dt → as x → ∞ and then (54) follows immediately. 48uppose now that g lies in C ∩ D p ∩ K p , with deg g = p − , for some p ∈ N ∗ . We first observe that ∆ px Z x + x Σg ( t ) dt = Z x + x ∆ p Σg ( t ) dt and that ∆ p Σg = c p + Σ∆ p g for some c p ∈ R . Since ∆ p g lies in C ∩ D ∩ K ,by (54) we have ∆ px R x + x Σg ( t ) dt∆ px Σg ( x + a ) = c p + R x + x Σ∆ p g ( t ) dtc p + Σ∆ p g ( x + a ) → as x → ∞ . Let us now show that the sequence n ∆ p − n R n + n Σg ( t ) dt∆ p − n Σg ( n + a ) (which exists for large values of n ) converges to . By minimality of p , thefunction ∆ p − Σg lies in D N \ D N and hence the sequence n ∆ p − Σg ( n + a ) tends to infinity. Using the discrete version of L’Hospital’s rule, we then obtain lim n → ∞ ∆ p − n R n + n Σg ( t ) dt∆ p − n Σg ( n + a ) = lim n → ∞ ∆ pn R n + n Σg ( t ) dt∆ pn Σg ( n + a ) = Iterating this process, we finally see that condition (53) holds for the integervalues of x , and then also for the real values of x using the eventual monotonicityof the function specified by (52) together with the squeeze theorem. Remark . In Proposition 6.26 we made the assumption that the functionspecified by (52) is eventually monotone. We conjecture that this assumptionis actually satisfied whenever g lies in ∪ p > ( C ∩ D p ∩ K p ) . The quest for a characterization of the domain of definition of the map Σ . Recall that the domain of definition of the map Σ is the set ∪ p > ( D p ∩ K p ) . Inthis respect, it would be useful to have a very simple test to check whether agiven function g lies in this set. The following result shows that both conditions g ∈ K and lim n → ∞ | g ( n + ) || g ( n ) | (55)are necessary. However, they are not sufficient. For instance, for any q ∈ N thefunction g q ( x ) = x q + + sin x lies in K q \ K q + and satisfies condition (55).However, it does not lie in D ∞ N . Proposition 6.28.
Let g ∈ K . If g lies in D p ∩ K p for some p ∈ N , thencondition (55) holds. Conversely, if condition (55) holds, then g lies in e D − N or we have ∆g ( x ) /g ( x ) → as x → ∞ . roof. Assume that g lies in D p ∩ K p for p = + deg g . If p = , then thefunction x | g ( x ) | eventually decreases to zero and hence condition (55) holds.Now suppose that p > . Then the function ∆ p g lies in D ∩ K and there aretwo exclusive cases to consider.(a) Suppose that the sequence n ∆ p − g ( n ) tends to infinity. Using thediscrete version of L’Hospital’s rule, we then obtain lim n → ∞ ∆ p − g ( n + ) ∆ p − g ( n ) = lim n → ∞ ∆ p g ( n + ) ∆ p g ( n ) = ℓ for some ℓ ∈ R satisfying | ℓ | . Iterating this process, we see thatcondition (55) holds.(b) Suppose that the sequence n ∆ p − g ( n ) has a nonzero limit. If p = ,then condition (55) holds trivially. If p > , then the sequence n ∆ p − g ( n ) tends to infinity and we can use the discrete version of L’Hospi-tal’s rule and iterate the process as in the previous case.Conversely, suppose that g ∈ K and that condition (55) holds. If theinequality is strict, then g is summable by the ratio test and hence g lies in e D − N . Otherwise, if the inequality is an equality, then we have | g ( n + ) | ∼ | g ( n ) | as n → N ∞ . Since g lies in K and hence eventually no longer changes insign (i.e., g lies in K − ), we also have g ( x + ) ∼ g ( x ) as x → ∞ , that is, ∆g ( x ) /g ( x ) → as x → ∞ .It is easy to see that the function g : R + → R lies in D ∞ N if and only if thereexists p ∈ N for which the sequence n ∆ p g ( n ) converges. This observationfollows from the immediate identity ∆ p g ( n ) = ∆ p g ( ) + n − X k = ∆ p + g ( k ) , n ∈ N ∗ , p ∈ N . In particular, if we assume that g ∈ K ∞ , then g does not lie to D ∞ N if and only iffor every p ∈ N the sequence n ∆ p g ( n ) tends to infinity. It is easy to see thatcondition (55) fails to hold for many functions g lying in K ∞ \ D ∞ N . Examples ofsuch functions include g ( x ) = x and g ( x ) = Γ ( x ) . It seems reasonable to thinkthat this observation actually follows from a general rule. We then formulatethe following conjecture. Conjecture.
Let g ∈ K ∞ . Then g lies in D ∞ N if and only if condition (55) holds. log Γ -type functions In this section we discuss certain differentiability properties of Σg when g liesin C r ∩ D p ∩ K p for some p , r ∈ N . In particular, when r p we show that Σg C r and that D r Σg ( x ) can be computed as the limit of the sequence n D r f pn [ g ]( x ) . We also discuss how the functions ( Σg ) ( r ) and Σg ( r ) arerelated and show how Σg can be computed by first computing Σg ( r ) . Finally,we provide an alternative uniqueness result for differentiable solutions to theequation ∆f = g . log Γ -type functions We investigate the differentiability of the function Σg when g is of class C r forsome r ∈ N . The central result is given in Theorem 7.2 below. We first considera technical lemma. For any n , r ∈ N , we set b rn ( x ) = D rx (cid:0) xn + r (cid:1) . Lemma 7.1.
For any integers r p , any g ∈ C r , and any a , x > wehave ρ p + a [ g ] ∈ C r and D rx ρ p + a [ g ]( x ) = ρ p + − ra [ g ( r ) ]( x )− r X i = p − r X j = b r + − ij ( x ) Z ρ p − j − r + a [ ∆ j + r − i g ( i ) ]( t ) dt . Proof.
It is clear that ρ p + a [ g ] ∈ C r and we can assume that r p . Using(5) it is then easy to see that D rx ρ p + a [ g ]( x ) = g ( r ) ( x + a ) − p − r X j = b rj ( x ) ∆ j + r g ( a ) ; D r − x ρ pa [ g ′ ]( x ) = g ( r ) ( x + a ) − p − r X n = b r − n ( x ) ∆ n + r − g ′ ( a ) . Subtracting the second equation from the first one, we obtain D rx ρ p + a [ g ]( x ) − D r − x ρ pa [ g ′ ]( x ) = p − r X n = b r − n ( x ) ∆ n + r − g ′ ( a )− p − r X j = b rj ( x ) ∆ ja ∆ r g ( a ) − p − j − r X n = G n ∆ n + r − g ′ ( a ) ! − p − r X j = b rj ( x ) ∆ ja p − j − r X n = G n ∆ n + r − g ′ ( a ) , where the expression in parentheses reduces to Z ρ p − j − r + a [ ∆ r − g ′ ]( t ) dt . p − r X n = b r − n ( x ) ∆ n + r − g ′ ( a ) − p − r X j = b rj ( x ) ∆ ja p − j − r X n = G n ∆ n + r − g ′ ( a )= p − r X n = b r − n ( x ) ∆ n + r − g ′ ( a ) − p − r X j = b rj ( x ) p − r X n = j G n − j ∆ n + r − g ′ ( a )= p − r X n = ∆ n + r − g ′ ( a ) b r − n ( x ) − n X j = G n − j b rj ( x ) . The latter expression in parentheses is identically zero. Indeed, the sum thereinis the convolution of the sequences n G n and n b rn ( x ) , whose ordinarygenerating functions are Z X n > (cid:0) tn (cid:1) z n dt = Z ( + z ) t dt = z ln ( + z ) and D rx X n > (cid:0) xn + r (cid:1) z n = D rx (cid:18) z r ( + z ) x (cid:19) = z r ( + z ) x ( ln ( + z )) r , respectively. Thus, the ordinary generating function for the convolution is z r − ( + z ) x ( ln ( + z )) r − and hence it defines the sequence n b r − n ( x ) .Now, collecting the remaining nonzero terms and using (21) we obtain D rx ρ p + a [ g ]( x ) = D r − x ρ pa [ g ′ ]( x ) − p − r X j = b rj ( x ) Z ρ p − j − r + a [ ∆ j + r − g ′ ]( t ) dt . Finally, using a simple induction on r , we obtain the claimed formula. Theorem 7.2 (Differentiability of multiple log Γ -type functions) . Let g ∈ C r ∩ D p ∩ K p for some r , p ∈ N . If r > p , we further assume that the derivatives g ( p + ) , g ( p + ) , . . . , g ( r ) lie in K . Then the following assertions hold.(a) Σg ∈ C r ∩ D p + ∩ K p .(b) For each x > , the sequence n D r f pn [ g ]( x ) converges and we have D r Σg ( x ) = lim n → ∞ D r f pn [ g ]( x ) , x > c) For any nonempty bounded subset E of R + , the sequence n D r f pn [ g ] converges uniformly on E to D r Σg .Proof. Let us first assume that r p . The result holds for r = by Theorem 3.4and Proposition 5.8. So we can assume that r p . Let x > and let m ∈ N ∗ be so that g is p -convex or p -concave on [ m , ∞ ) . For every i ∈ {
1, . . . , r } andevery j ∈ {
0, . . . , p − r } , by Lemma 2.2(b), Lemma 3.3, Proposition 4.8, andLemma 6.16, both sequences n n − X k = m ρ p + − rk [ g ( r ) ]( x ) and n n − X k = m Z ρ p − j − r + k [ ∆ j + r − i g ( i ) ]( t ) dt converge and, for any integer n > m , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k = n ρ p + − rk [ g ( r ) ]( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ⌈ x ⌉ (cid:12)(cid:12)(cid:12)(cid:0) x − p − r (cid:1)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∆ p − r g ( r ) ( n ) (cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k = n Z ρ p − j − r + k [ ∆ j + r − i g ( i ) ]( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G p − j − r (cid:12)(cid:12)(cid:12) ∆ p − i g ( i ) ( n ) (cid:12)(cid:12)(cid:12) . Combining these inequalities with Lemma 7.1, it follows that for any boundedsubset E of R + the sequence n sup x ∈ E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k = n D rx ρ p + k [ g ]( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) converges to zero. Using the classical result on differentiability of uniformlyconvergent sequences, it follows that the function ∞ X k = m ρ p + k [ g ]( x ) = f pm [ g ]( x ) − Σg ( x ) lies in C r (and hence so does Σg ) and that ∞ X k = m D r ρ p + k [ g ]( x ) = D r ∞ X k = m ρ p + k [ g ]( x ) = D r f pm [ g ]( x ) − D r Σg ( x ) . This proves the theorem when r p .Let us now assume that r > p . By Proposition 4.8, the function g ( p ) lies in C r − p ∩ D ∩ K . By Proposition 4.11, for any i ∈ { p +
1, . . . , r } , the function g ( i ) lies in C r − i ∩ e D − N ∩ K . By Theorem 3.7, it follows that the sequence n − n − X k = g ( i ) ( x + k ) = D i f pn [ g ]( x ) converges uniformly on R + . Again, we conclude the proof by using the classicalresult on differentiability of uniformly convergent sequences.53 emark . If g ∈ C r ∩ D p ∩ K p for some integers r p , then the function Σg lies in C r by Theorem 7.2. Actually, this result can also be established byelementary means. Indeed, by Proposition 5.8 we have Σg ( r ) ∈ C . Hence, thereexists F ∈ C r such that F ( r ) = Σg ( r ) . Since Σg ( r ) also lies in K p − r , we have F ∈ K p by Proposition 4.8. Now, we also have D r ∆F = ∆F ( r ) = ∆Σg ( r ) = g ( r ) ,which shows that ∆ ( F + P ) = g for some polynomial P of degree at most r . Since F + P lies in K p , by the uniqueness theorem we must have F + P = Σg + c forsome c ∈ R . Hence Σg lies in C r . Proposition 7.4.
Let g ∈ C r ∩ D p ∩ K p for some integers p ∈ N and r ∈ N ∗ .If r > p , we further assume that the derivatives g ( p + ) , g ( p + ) , . . . , g ( r ) liein K . Then for any x > we have ( Σg ) ( r ) ( x ) − Σg ( r ) ( x ) = ( Σg ) ( r ) ( ) = g ( r − ) ( ) − σ [ g ( r ) ] . (56) If r > p , then this value reduces to − P ∞ k = g ( r ) ( k ) .Proof. By Propositions 4.8 and 4.11, we have g ( r ) ∈ D ( p − r ) + N . By Theorem 7.2,we have Σg ∈ C r ∩ D p + ∩ K p . Also, by the existence Theorem 3.4, bothfunctions ϕ = ( Σg ) ( r ) and ϕ = Σg ( r ) are solutions in K ( p − r ) + to the equation ∆ϕ = g ( r ) . By the uniqueness Theorem 3.1, we have ( Σg ) ( r ) − Σg ( r ) = c forsome c ∈ R . For any x > , using (37) we then get g ( r − ) ( ) − σ [ g ( r ) ] = g ( r − ) ( x ) − Z x + x Σg ( r ) ( t ) dt = c + g ( r − ) ( x ) − Z x + x ( Σg ) ( r ) ( t ) dt = c + g ( r − ) ( x ) − ( Σg ) ( r − ) ( x + ) + ( Σg ) ( r − ) ( x ) , which reduces to the constant c . If r > p , then we have g ( r ) ∈ C ∩ e D − N ∩ K byProposition 4.11. The last part of the proof then follows from Proposition 6.25. Example 7.5.
The function g ( x ) = x lies in C ∞ ∩ D ∩ K ∞ and all its derivativeslie in K . By Theorem 7.2, the function Σg ( x ) = ∞ X k = (cid:18) k + − x + k (cid:19) = ψ ( x ) + γ lies in C ∞ ∩ D ∩ K ∞ . Thus, the series can be differentiated term by term andhence, for any r ∈ N ∗ , we have ( Σg ) ( r ) ( x ) = − ∞ X k = (− ) r r ! ( x + k ) − r − = ψ r ( x ) . ( Σg ) ( r ) . Corollary 7.6.
Let g ∈ C r ∩ D p ∩ K p for some p , r ∈ N and let x > beso that g is p -convex or p -concave on [ x , ∞ ) . If r > p , we further assumethat the derivatives g ( p + ) , g ( p + ) , . . . , g ( r ) lie in K . Then the followingassertions hold.(a) For any a > , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( Σg ) ( r ) ( x + a ) − ( Σg ) ( r ) ( x ) − ( p − r ) + X j = (cid:0) aj (cid:1) ∆ j − g ( r ) ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ⌈ a ⌉ (cid:12)(cid:12)(cid:12)(cid:0) a − ( p − r ) + (cid:1)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∆ ( p − r ) + g ( r ) ( x ) (cid:12)(cid:12)(cid:12) In particular, the left-hand expression tends to zero as x → ∞ .(b) If r > , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( Σg ) ( r ) ( x ) − g ( r − ) ( x ) + ( p − r ) + X j = G j ∆ j − g ( r ) ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G ( p − r ) + | ∆ ( p − r ) + g ( r ) ( x ) | . In particular, the left-hand expression tends to zero as x → ∞ .Moreover, if r > p , then ( Σg ) ( r ) ( x ) → as x → ∞ . It turns out that the convergence results in Corollary 7.6 can be obtainedjust by taking the r th derivative of (33) and (40), respectively. In particular,the function J p + [ Σg ]( x ) and its derivatives vanish at infinity. Derivatives of Σg ( x ) at x = . Proposition 7.4 enables us to compute thevalue of ( Σg ) ( r ) ( ) whenever g ( r ) exists. For instance, for g ( x ) = ln x we obtain ψ ( ) = ( ln Γ ) ′ ( ) = − σ [ g ′ ] = − γ and, for any integer r > , ψ ( r − ) ( ) = ( ln Γ ) ( r ) ( ) = (− ) r ( r − ) ! − σ [ g ( r ) ]= (− ) r ( r − ) ! ζ ( r ) . If the function Σg is real analytic at , then the following Taylor series expansion Σg ( x + ) = ∞ X k = ( Σg ) ( k ) ( ) x k k ! (57)55olds in some neighborhood of x = . For instance, for g ( x ) = ln x we obtain ln Γ ( x + ) = − γx + ∞ X k = (− ) k ζ ( k ) k x k , | x | < Exponential generating function for the sequence σ [ g ( n ) ] . Suppose that g lies in C ∞ ∩ D p ∩ K p for some p ∈ N and that g ( k ) lies in K for any k ∈ N .Identity (56) enables us to write formally the following power series ∞ X k = σ [ g ( k ) ] x k k ! = σ [ g ] + Z x + g ( t ) dt − Σg ( x + ) . (58)Thus, the right side of (58) is precisely the exponential generating function egf σ [ g ]( x ) for the sequence n σ [ g ( n ) ] . We then have egf σ [ g ]( x ) = − J [ Σg ]( x + ) , x > and hence σ [ g ( k ) ] = ( egf σ [ g ]) ( k ) ( ) = − ( J [ Σg ]) ( k ) ( ) , k ∈ N . For instance, if g ( x ) = ln x , then we have σ [ g ] = − +
12 ln ( π ) , σ [ g ′ ] = γ ,and for any integer k > σ [ g ( k ) ] = (− ) k ( k − ) ! ( − ( k − ) ζ ( k )) . Similarly, if the sequence n σ [ g ( n ) ] is defined, then the correspondingexponential generating function egf σ [ g ]( x ) is ∞ X k = σ [ g ( k ) ] x k k ! = σ [ g ] + Z x g ( t ) dt − Σg ( x + ) . Now, if p = + deg g , then by Propositions 4.8, 6.23, and 6.25 we also have σ [ g ( k ) ] = γ [ g ( k ) ] + ( p − k ) + X j = G j ∆ j − g ( k ) ( ) , k ∈ N and hence the exponential generating function for the sequence n γ [ g ( n ) ] isthe function egf γ [ g ]( x ) = egf σ [ g ]( x ) − p X j = G j p − j X k = x k k ! ∆ j − g ( k ) ( ) . Analogues of Euler’s series representation of γ . Integrating both sides of(57) on (
0, 1 ) (assuming that the series can be integrated term by term), weobtain the identity σ [ g ] = ∞ X k = ( Σg ) ( k ) ( )( k + ) ! . (59)56imilarly, integrating both sides of (58) on (
0, 1 ) (assuming that the series canbe integrated term by term), we obtain the identity ∞ X k = σ [ g ( k ) ]( k + ) ! = Z ( − t ) g ( t ) dt . (60)Taking for instance g ( x ) = /x in (59), we immediately retrieve Euler’s seriesrepresentation of γ (see, e.g., [76, p. 272]) γ = ∞ X k = (− ) k ζ ( k ) k . This formula can also be obtained by taking g ( x ) = /x in (60) and using thestraightforward identity σ [ g ( k ) ] = (− ) k k ! (cid:18) ζ ( k + ) − k (cid:19) , k ∈ N ∗ . Considering different functions g ( x ) in (59) and (60) enables one to derivevarious interesting identities. Example 7.7.
Taking g ( x ) = ψ ( x ) in (60) and using the straightforward iden-tity σ [ g ( k ) ] = (− ) k − ( k − )( k − ) ! ζ ( k ) k ∈ N , k > we obtain ∞ X k = (− ) k k − k ( k + ) ζ ( k ) = − ln ( π ) . Similarly, taking g ( x ) = ln x and then g ( x ) = ln Γ ( x ) in (59) and (60) we obtainthe identities ∞ X k = (− ) k k ( k + ) ζ ( k ) = γ − +
12 ln ( π ) , ∞ X k = (− ) k ( k + )( k + ) ζ ( k ) = + γ − A , ∞ X k = (− ) k k − k ( k + )( k + ) ζ ( k ) = −
14 ln ( π ) − A , where A is Glaisher-Kinkelin’s constant; see also [76, Section 3.4]. Given r ∈ N ∗ and a function g ∈ C r , a solution f ∈ C r to the equation ∆f = g can sometimes be found by first searching for an appropriate solution ϕ ∈ C
57o the equation ∆ϕ = g ( r ) and then calculating f as an r th antiderivative of ϕ .To our knowledge, this approach was investigated thoroughly by Krull [47] andthen by Dufresnoy and Pisot [30]. Here we present a general theory based onthis idea.We first observe that if ϕ ∈ C is a solution to the equation ∆ϕ = g ( r ) , thenthe map x Z x + x ϕ ( t ) dt − g ( r − ) ( x ) has a zero derivative and hence it is constant on R + . In particular, it has a finiteright limit at x = . Recall also that the Bernoulli numbers B B B
2, . . . aredefined implicitly by the single equation (see, e.g., [34, p. 284]) m X j = (cid:0) m + j (cid:1) B j = m , integer m > Theorem 7.8.
Let r ∈ N ∗ , a > , g ∈ C r , and let ϕ : R + → R be a continuoussolution to the equation ∆ϕ = g ( r ) . Then there exists a solution f ∈ C r tothe equation ∆f = g such that f ( r ) = ϕ if and only if Z a + a ϕ ( t ) dt = g ( r − ) ( a ) . (61) If any of these equivalent conditions holds, then f is uniquely determined(up to an additive constant) by f ( x ) = f ( a ) + r − X k = c k ( x − a ) k k ! + Z xa ( x − t ) r − ( r − ) ! ϕ ( t ) dt , (62) where, for k =
1, . . . , r − , c k = r − k − X j = B j j ! (cid:18) g ( j + k − ) ( a ) − Z a + a ( a + − t ) r − j − k ( r − j − k ) ! ϕ ( t ) dt (cid:19) . (63) Proof.
Condition (61) is clearly necessary. Indeed, we have Z a + a ϕ ( t ) dt = f ( r − ) ( a + ) − f ( r − ) ( a ) = g ( r − ) ( a ) . Let us show that it is sufficient. Since ϕ is continuous, there exists f ∈ C r such that f ( r ) = ϕ . Taylor’s theorem then provides the expansion formula (62)with arbitrary parameters c k = f ( k ) ( a ) for k =
1, . . . , r − . Now we needto determine the parameters c
1, . . . , c k for f to be a solution to the equation ∆f = g . To this extent, we need the following claim. Claim.
The function f satisfies the equation ∆f = g if and only if f ( r ) satisfiesthe equation ∆f ( r ) = g ( r ) and ∆f ( j ) ( a ) = g ( j ) ( a ) for j =
0, . . . , r − .58 roof of the claim. The condition is clearly necessary. To see that it is suffi-cient, we simply show by decreasing induction on j that ∆f ( j ) = g ( j ) . Clearly,this is true for j = r . Suppose that it is true for some integer j satisfying j r . For any x > we have ∆f ( j − ) ( x ) − ∆f ( j − ) ( a ) = Z xa ∆f ( j ) ( t ) dt = Z xa g ( j ) ( t ) dt = g ( j − ) ( x ) − g ( j − ) ( a ) = g ( j − ) ( x ) − ∆f ( j − ) ( a ) , which shows that the result still holds for j − .By the claim, f satisfies the equation ∆f = g if and only if ∆f ( j ) ( a ) = g ( j ) ( a ) for j =
0, . . . , r − . When j = r − , the latter condition is nothing other thancondition (61) and hence it is satisfied. Applying Taylor’s theorem to f ( j ) , weobtain f ( j ) ( a + ) − f ( j ) ( a ) = r − j − X k = k ! f ( j + k ) ( a ) + Z a + a ( a + − t ) r − j − ( r − j − ) ! ϕ ( t ) dt , and hence we see that the remaining r − conditions are r − j − X k = k ! c j + k = d j , j =
0, . . . , r − where d j = g ( j ) ( a ) − Z a + a ( a + − t ) r − j − ( r − j − ) ! ϕ ( t ) dt , j =
0, . . . , r − c k = f ( k ) ( a ) , k =
1, . . . , r − It is not difficult to see that these r − conditions form a consistent triangularsystem of r − linear equations in the r − unknowns c
1, . . . , c r − . Thisestablishes the uniqueness of f up to an additive constant.Let us now show that formula (63) holds. For k =
1, . . . , r − , we have r − k − X j = B j j ! d j + k − = r − k − X j = B j j ! r − j − k X i = i ! c i + j + k − Replacing i by i − j − k + and then permuting the resulting sums, the latterexpression reduces to r − k − X j = B j j ! r − X i = j + k ( i − j − k + ) ! c i = r − X i = k c i ( i − k + ) ! i − k X j = (cid:0) i − k + j (cid:1) B j = r − X i = k c i ( i − k + ) ! 0 i − k = c k . This completes the proof of the theorem.59dding an appropriate constant to ϕ if necessary in Theorem 7.8, we canalways assume that condition (61) holds. More precisely, the function ϕ ⋆ = ϕ + C , where C = g ( r − ) ( a ) − Z a + a ϕ ( t ) dt , satisfies R a + a ϕ ⋆ ( t ) dt = g ( r − ) ( a ) . In fact, this is exactly what we did inProposition 7.4, where (56) represents the equation f ( r ) ( x ) − ϕ ( x ) = g ( r − ) ( ) − Z ϕ ( t ) dt . Example 7.9.
Let g ∈ C , let G ∈ C be defined by the equation G ( x ) = Z x g ( t ) dt , and let f ∈ C be any solution to the equation ∆f = g . Then the function F ∈ C defined by the equation F ( x ) = Z x f ( t ) dt − ( x − ) Z f ( t ) dt , is a solution to the equation ∆F = G . Moreover, if f ∈ K p for some p ∈ N ,then F ∈ K p + by Lemma 2.2(g). For similar results, see [47, p. 254] and [50,Section 2].Now, using Theorem 7.2, Proposition 7.4, and Theorem 7.8, we can easilyderive the following useful corollary. Corollary 7.10.
Let g ∈ C r ∩ D p ∩ K p for some p ∈ N and some r ∈ N ∗ . If r > p , we further assume that the derivatives g ( p + ) , g ( p + ) , . . . , g ( r ) lie in K . Then Σg ∈ C r ∩ D p + ∩ K p and ( Σg ) ( r ) − Σg ( r ) = g ( r − ) ( ) − σ [ g ( r ) ] . (This value reduces to − P ∞ k = g ( r ) ( k ) if r > p .) Moreover, for any a > ,we have Σg = f a − f a ( ) , where f a ∈ C r is defined by f a ( x ) = r − X k = c k ( a ) ( x − a ) k k ! + Z xa ( x − t ) r − ( r − ) ! ( Σg ) ( r ) ( t ) dt and, for k =
1, . . . , r − , c k ( a ) = r − k − X j = B j j ! (cid:18) g ( j + k − ) ( a ) − Z a + a ( a + − t ) r − j − k ( r − j − k ) ! ( Σg ) ( r ) ( t ) dt (cid:19) . Σg from an explicit expression for Σg ( r ) . The followingtwo examples illustrate the use of this result. Example 7.11.
The function g ( x ) = R x ( x − t ) ln t dt lies in C ∞ ∩ D ∩ K ∞ .Choosing r = and a = (as a limiting value) in Corollary 7.10, we get g ′′ ( x ) = ln x , Σg ′′ ( x ) = ln Γ ( x ) , ( Σg ) ′′ ( x ) = ln Γ ( x ) −
12 ln ( π ) , and Σg ( x ) = − ( ln A ) x −
14 ln ( π ) x + Z x ( x − t ) ln Γ ( t ) dt , where A is Glaisher-Kinkelin’s constant and the integral is the polygammafunction ψ − ( x ) . Using Theorem 6.5, we also obtain the following asymptoticbehavior of ΣgΣg ( x ) + ( x − x + x ) − x ( x − ) ln x − ( x + ) ( x + ) + ( x + ) ( x + ) → ζ ( ) π as x → ∞ . Example 7.12.
The function g ( x ) = ar tan ( x ) lies in C ∞ ∩ D ∩ K ∞ . Choosing r = and a = (as a limiting value) in Corollary 7.10, we get g ′ ( x ) = ( x + ) − = − ℑ ( x + i ) − Σg ′ ( x ) = ℑ ψ ( + i ) − ℑ ψ ( x + i ) , ( Σg ) ′ ( x ) = π − ℑ ψ ( x + i ) , and Σg ( x ) = π ( x − ) + ℑ ln Γ ( + i ) − ℑ ln Γ ( x + i )= c + π x + i Γ ( x + i ) Γ ( x − i ) for some c ∈ R . Using Theorem 6.5, we also obtain the inequality (cid:12)(cid:12)(cid:12)(cid:12) Σg ( x ) − (cid:18) x − (cid:19) ar tan ( x ) +
12 ln ( x + ) − + π − ℑ ln Γ ( + i ) (cid:12)(cid:12)(cid:12)(cid:12)
12 ar tan 1 x + x + and hence the left side approaches zero as x → ∞ .61 .3 An alternative uniqueness result The following theorem provides a uniqueness result for differentiable solutionsto the equation ∆f = g . These solutions can be computed from their derivativesusing Theorem 7.8. Fact 7.13.
A periodic function ω : R + → R is constant if and only if it liesin K . Theorem 7.14.
Let r ∈ N ∗ and g ∈ C r , and assume that there exists ϕ ∈ C r such that ∆ϕ = g and ϕ ( r ) ∈ R N . Then, the following assertions hold.(i) For each x > , the series ∞ X k = g ( r ) ( x + k ) converges.(ii) For any f ∈ C r ∩ K r − such that ∆f = g , we have f = c + ϕ for some c ∈ R and f ( r ) ( x ) = − ∞ X k = g ( r ) ( x + k ) . Proof.
Assertion (i) follows immediately from (12) and we clearly have ϕ ( r ) ( x ) = − ∞ X k = g ( r ) ( x + k ) , x > Now, let f ∈ C r ∩ K r − be such that ∆f = g . Negating f , ϕ , and g if necessary,we can assume that f ∈ K r − + , which implies that f ( r ) is eventually nonnegativeby Lemma 2.2(e). To complete the proof, by Proposition 4.4 and Fact 7.13 itis enough to show that the -periodic function ω = f − ϕ lies in K r − , i.e., itsatisfies ω ( r ) > on R + . Suppose on the contrary that ω ( r ) ( z ) < for some z > . Since ω is -periodic, we have f ( r ) ( z + m ) < ϕ ( r ) ( z + m ) , for large integer m . In particular, we have < − ω ( r ) ( z ) = − ω ( r ) ( z + m ) ϕ ( r ) ( z + m ) for large integer m , which contradicts the assumption that ϕ ( r ) ∈ R N . Thisproves assertion (ii). Example 7.15.
The assumptions of Theorem 7.14 hold if g ( x ) = ln x , ϕ ( x ) = ln Γ ( x ) , and r = . It follows from Theorem 7.14 that all solutions to theequation ∆f = g that lie in C ∩ K are of the form f ( x ) = c + ln Γ ( x ) , where c ∈ R . We thus retrieve the Bohr-Mollerup-Artin Theorem with the additionalassumption that f lies in C . 62 Further results
Keeping in mind the objective of generalizing Webster’s formulas to multiple log Γ -type functions, we now explore further questions related to our mainresults. In particular, we provide for multiple log Γ -type functions analoguesof Euler’s infinite product , Weierstrass’ infinite product , Raabe’s formula , Gauss’ multiplication formula , and
Wallis’s product formula . Let g ∈ D p ∩ K p for some p ∈ N . As we already observed in the Introduction,the representation of Σg as the pointwise limit of the sequence n f pn [ g ] is theanalogue of Gauss’ limit for the gamma function. Using identity (18), we cansee that this form of Σg can be easily translated into a series, namely Σg ( x ) = f p [ g ]( x ) − ∞ X k = ρ p + k [ g ]( x ) , x > We also observe that, when g ( x ) = ln x and p = , the multiplicative versionof this series representation reduces to the classical Euler product form of thegamma function (see, e.g., [76, p. 3]), as given in Example 8.2 below. Thus, forany multiple log Γ -type function, the series representation above is the analogueof the Eulerian form of the gamma function. Rewriting this identity explicitlyand using the uniform convergence of the sequence n f pn [ g ] (cf. Theorem 3.4),we immediately obtain the following result. Theorem 8.1 (Eulerian form) . Let g ∈ D p ∩ K p for some p ∈ N . Then Σg ( x ) = − g ( x ) + p X j = (cid:0) xj (cid:1) ∆ j − g ( ) − ∞ X k = g ( x + k ) − p X j = (cid:0) xj (cid:1) ∆ j g ( k ) and the series converges uniformly on any bounded subset of R + . Recall also that the uniform convergence enables one to integrate the seriesabove term by term on any bounded interval (see Proposition 5.8). We can alsodifferentiate the series term by term as shown in Theorem 7.2.
Example 8.2.
Considering the function g ( x ) = ln x for which p = + deg g = ,we obtain the following infinite product representations for any x > : Γ ( x ) = x ∞ Y k = ( + /k ) x + x/k , e ψ ( x ) = e − x ∞ Y k = ( + /k ) e − x + k , and e R x Γ ( t ) dt = e x x x ∞ Y k = e x ( + /k ) x / ( + x/k ) x + k . .2 Weierstrassian form We now show that the classical Weierstrass factorization of the gamma function(see Example 8.6 below) can be generalized to any log Γ p -type function that isof class C p . The following two theorems deal separately with the cases p = and p > . Note that the case p = was previously established by John [42,Theorem B’] and in the multiplicative notation by Webster [80, Theorem 7.1]. Theorem 8.3 (Weierstrassian form) . For any g ∈ C ∩ D ∩ K , we have γ [ g ] = σ [ g ] and Σg ( x ) = σ [ g ] − g ( x ) − ∞ X k = (cid:18) g ( x + k ) − Z k + k g ( t ) dt (cid:19) and the series converges uniformly on any bounded subset of R + .Proof. The result immediately follows from Proposition 6.23, Eq. (48), andTheorem 8.1.
Lemma 8.4.
Let g ∈ C ∩ D p ∩ K p for some p ∈ N ∗ . Then ∆g ( x ) − p − X j = G j ∆ j g ′ ( x ) → as x → ∞ . If, in addition, g ∈ C p − , then ∆ p − g ( x ) − g ( p − ) ( x ) → as x → ∞ .Proof. The first convergence result follows immediately from (41). Let us nowassume that g ∈ C p − . For every i ∈ {
0, . . . , p − } , the function g i = ∆ i g ( p − − i ) lies in C ∩ D ∩ K and hence, using the first result, we see that ∆g i ( x )− g ′ i ( x ) → as x → ∞ . Summing these limits for i =
0, . . . , p − , we obtain the claimedlimit. Theorem 8.5 (Weierstrassian form) . Let g ∈ C p ∩ D p ∩ K p with deg g = p − for some p ∈ N ∗ . Then we have γ [ g ( p ) ] = σ [ g ( p ) ] = g ( p − ) ( ) − ( Σg ) ( p ) ( ) and Σg ( x ) = p − X j = (cid:0) xj (cid:1) ∆ j − g ( ) + (cid:0) xp (cid:1) ( Σg ) ( p ) ( )− g ( x ) − ∞ X k = g ( x + k ) − p − X j = (cid:0) xj (cid:1) ∆ j g ( k ) − (cid:0) xp (cid:1) g ( p ) ( k ) and the series converges uniformly on any bounded subset of R + . roof. The identities involving the constants follow from Propositions 4.8, 6.23,and 7.4. Now, using (48) we get γ [ g ( p ) ] = ∞ X k = ( g ( p ) ( k ) − ∆g ( p − ) ( k )) . Using Theorem 8.1, we then obtain Σg ( x ) = p − X j = (cid:0) xj (cid:1) ∆ j − g ( ) + (cid:0) xp (cid:1) (cid:16) g ( p − ) ( ) − γ [ g ( p ) ] (cid:17) − g ( x ) − lim n → ∞ n − X k = g ( x + k ) − p − X j = (cid:0) xj (cid:1) ∆ j g ( k ) − (cid:0) xp (cid:1) g ( p ) ( k ) + lim n → ∞ (cid:0) xp (cid:1) (cid:16) ∆ p − g ( n ) − g ( p − ) ( n ) (cid:17) , where the latter limit is zero by Lemma 8.4. Also, the uniform convergence isensured by Theorem 8.1.It is important to note that, just as the series given in Theorem 8.1, theseries given in Theorems 8.3 and 8.5 also represent the limit of the sequence n f pn [ g ]( x ) . Thus, by Theorem 7.2, those series can be integrated and differentiatedterm by term. Example 8.6.
Considering the function g ( x ) = ln x for which p = + deg g = , we retrieve the following Weierstrassian form of the gamma function in aneffortless way Γ ( x ) = e − γx x ∞ Y k = e xk + xk , x > Remark . Under the assumptions of Lemma 8.4, by Propositions 4.5 and4.8 we have g ′ ∈ R p − R , i.e., for any a > g ′ ( x + a ) − p − X j = (cid:0) aj (cid:1) ∆ j g ′ ( x ) → as x → ∞ . Combining this with Lemma 8.4, we can derive surprising limits. For instance,if p ∈ {
1, 2, 3 } , then ∆g ( x ) − g ′ ( x + ) → as x → ∞ . Recall that Raabe’s formula yields, for any x > , a simple explicit expressionfor the integral of the log-gamma function over the interval ( x , x + ) . That is, Z x + x ln Γ ( t ) dt =
12 ln ( π ) + x ln x − x , x > (64)65n particular, setting x = , we obtain the identity Z
21 ln Γ ( t ) dt = − +
12 ln ( π ) , which is precisely the value of σ [ g ] when g ( x ) = ln x . For recent references onRaabe’s formula, see, e.g., [27] and see [76, p. 29].Clearly, identities (36) and (37) enable us to define for any continuous mul-tiple log Γ -type function Σg the analogue of Raabe’s formula. Thus defined, thisnew formula can be obtained simply by computing the value σ [ g ] , or even thevalue σ [ g ] (see Definition 6.14) when g is integrable at .In general, the value of σ [ g ] can be computed using Proposition 5.8(c2).Specifically, if g ∈ C ∩ D p ∩ K p for some p ∈ N , we have σ [ g ] = lim n → ∞ Z ( f pn [ g ]( t ) + g ( t )) dt = lim n → ∞ n − X k = g ( k ) − Z n g ( t ) dt + p X j = G j ∆ j − g ( n ) . (65)which is nothing other than the restriction of the generalized Stirling formula(40) to the natural integers. Equivalently, this value can be obtained by in-tegrating on the interval (
0, 1 ) the series representation of Σg + g given inTheorem 8.1. That is, σ [ g ] = p X j = G j ∆ j − g ( ) − ∞ X k = Z k + k g ( t ) dt − p X j = G j ∆ j g ( k ) . (66)Note also that, under certain assumptions, the series above converges to zero as p → N ∞ ; see Proposition 6.8. Example 8.8. If g ( x ) = x , we obtain σ [ g ] = ∞ X k = (cid:18) k − ln (cid:18) + k (cid:19)(cid:19) , which is the Euler constant γ . Identity (37) then immediately provides thefollowing analogue of Raabe’s formula Z x + x ψ ( t ) dt = ln x , x > Webster [80, Theorem 5.2] showed how an analogue of Gauss’ multiplicationformula can be constructed for any Γ -type function. His proof is very easy and66ssentially uses the uniqueness theorem. We now show that this formula canbe further extended to multiple Γ -type functions. As usual, we use the additivenotation. Theorem 8.9 (Gauss’ multiplication formula) . Let p ∈ N , m ∈ N ∗ , and g ∈ D p ∩ K p . Define also the functions g m , h m : R + → R by the equations g m ( x ) = g ( xm ) and h m ( x ) = g ( x ) − g m ( x ) for x > . Then g m ∈ D p ∩ K p and, for x > , m − X j = ( Σg ) (cid:18) x + jm (cid:19) = m − X j = ( Σg ) (cid:18) jm (cid:19) + Σg m ( x ) . (67) If h m ∈ D p ∩ K p , then for x > , m − X j = ( Σg ) (cid:18) x + jm (cid:19) + Σh m ( x ) = m − X j = ( Σg ) (cid:18) jm (cid:19) + Σg ( x ) . Proof.
We clearly have g m ∈ K p . Also, it is easy to see that g m ∈ D p ∩ K p (we can use Proposition 4.15 for instance). Now, we can readily check that thefunction f : R + → R defined by f ( x ) = m − X j = ( Σg ) (cid:18) x + jm (cid:19) − m − X j = ( Σg ) (cid:18) jm (cid:19) is a solution to the equation ∆f = g m that lies in K p and such that f ( ) = . Bythe uniqueness Theorem 3.1, it follows that f = Σg m . Now, if h m ∈ D p ∩ K p ,then we also have f = Σg − Σh m . Corollary 8.10.
Let g ∈ C ∩ D p ∩ K p for some p ∈ N . Define also thefunctions g m : R + → R ( m ∈ N ∗ ) by the equation g m ( x ) = g ( xm ) . Then wehave lim m → ∞ ( Σg m )( mx ) − ( Σg m )( m ) m = Z x g ( t ) dt , x > Moreover, if g is integrable at , then lim m → ∞ m ( Σg m )( mx ) = Z x g ( t ) dt , x > Proof.
Replacing x by mx in (67) and dividing through by m , we obtain twoRiemann sums that converge, letting m → N ∞ , to the integrals of Σg ( x + t ) and Σg ( t ) over t ∈ (
0, 1 ) . Combining the resulting equation with (37) gives theresult.To use Theorem 8.9 to its full capacity, a closed-form expression for theright-hand sum of identity (67) would be welcome. The following proposition67rings a partial answer to this natural question. Recall first that B k denotesthe k th Bernoulli number (see Subsection 7.2). Also, for n , r ∈ N , b rn ( x ) standsfor the function D rx (cid:0) xn + r (cid:1) . Finally, ∆ [ h ] denotes the forward difference operatorwith step h . Lemma 8.11.
For any m , q ∈ N ∗ , we have m X i = (cid:0) i/mq (cid:1) = m G q + (cid:0) q (cid:1) + q + X i = B i i ! m i − (cid:16) b i − q − i + ( ) − b i − q − i + ( ) (cid:17) = m G q + (cid:0) q (cid:1) − q + X i = G i (cid:16) ( ∆ i − [ m ] b q )( ) − ( ∆ i − [ m ] b q )( ) (cid:17) . Also, for i =
1, . . . , q + , we have b i − q − i + ( ) − b i − q − i + ( ) = q ! q X k = i (− ) k − q (cid:20) qk (cid:21) k ! ( k − i + ) ! , where (cid:2) qk (cid:3) is the number of ways to arrange q objects into k cycles (Stirlingnumber of the first kind).Proof. The first formula results from a straightforward application of the Euler-Maclaurin formula (Proposition 6.21) with a = , b = , and N = m . Weprove the second formula similarly using the general form of Gregory’s formula(Proposition 6.20) with a = , n = m , and h = m . The last part follows fromthe classical linear decomposition of binomial coefficients into ordinary powers(see, e.g., [34, p. 263]). Proposition 8.12.
Let p ∈ N , m ∈ N ∗ , g ∈ C ∩ D p ∩ K p , and set c m , j = m X i = (cid:0) i/mj (cid:1) , for j =
1, . . . , p . Then m X j = ( Σg ) (cid:18) jm (cid:19) = m σ [ g ] − σ [ g m ]+ lim n → ∞ (cid:18) m Z n g ( t ) dt − Z mn + g m ( t ) dt + p X j = (cid:0) ( c m , j − m G j ) ∆ j − g ( n ) + G j ∆ j − g m ( mn + ) (cid:1) (cid:19) , where the numbers c m , j can be computed using Lemma 8.11 and the func-tion g m : R + → R is defined by the equation g m ( x ) = g ( xm ) . roof. We have m X j = ( Σg ) (cid:18) jm (cid:19) = lim n → ∞ m X j = f pn [ g ] (cid:18) jm (cid:19) = − m X j = g (cid:18) jm (cid:19) + lim n → ∞ n − X k = m g ( k ) − m X j = g (cid:18) jm + k (cid:19) + p X j = c m , j ∆ j − g ( n ) . Also, n − X k = m X j = g (cid:18) jm + k (cid:19) = n − X k = ∆ k km X j = g (cid:18) jm (cid:19) = mn X j = g (cid:18) jm (cid:19) − m X j = g (cid:18) jm (cid:19) . Thus, we have m X j = ( Σg ) (cid:18) jm (cid:19) = lim n → ∞ m n − X k = g ( k ) − mn X k = g (cid:18) km (cid:19) + p X j = c m , j ∆ j − g ( n ) . The claimed formula then follows from identity (65).
Example 8.13.
Let us apply Theorem 8.9 to the function g ( x ) = ln x . We have g m ( x ) = ln x − ln m and Σg m ( x ) = ln Γ ( x ) − ( x − ) ln m . Hence, we retrievethe following Gauss multiplication formula m − Y j = Γ (cid:18) x + jm (cid:19) = Γ ( x ) m x − m − Y j = Γ (cid:18) jm (cid:19) , x > and it can be proved using Proposition 8.12 that the right-hand product is m − ( π ) m −
12 .
When m = , this identity reduces to Legendre’s duplication formula Γ (cid:16) x (cid:17) Γ (cid:18) x + (cid:19) = Γ ( x ) x − √ π , x > The following result provides an asymptotic expansion of the left-hand sumof identity (67). This expansion can be used for instance to estimate the integral(37) (and hence also the asymptotic constant σ [ g ] ), e.g, using Richardson’sextrapolation method. As a byproduct, this result also provides an asymptoticexpansion of Σg (or even of the difference between Σg and its trend) in termsof the higher derivatives of g . We omit the proof for it is a straightforwardapplication of Euler-Maclaurin’s formula (Proposition 6.21) with a = , b = ,and N = m . 69 roposition 8.14. Let g ∈ C q ∩ D p ∩ K p for some p ∈ N and some q ∈ N ∗ .Then, for any m ∈ N ∗ and any x > , we have m m − X j = ( Σg ) (cid:18) x + jm (cid:19) = Z x + x Σg ( t ) dt − m g ( x ) + q X k = m k B k ( k ) ! D k − g ( x ) + R m , q ( x ) , with | R m , q ( x ) | m q | B q | ( q ) ! Z x + x | D q Σg ( t ) | dt . In particular, Σg ( x ) = σ [ g ] + Z x g ( t ) dt − g ( x ) + q X k = B k ( k ) ! D k − g ( x ) + R q ( x ) . Example 8.15.
Taking g ( x ) = ln x in the second part of Proposition 8.14,for any q ∈ N ∗ we obtain the following asymptotic expansion as x → ∞ (see,e.g., [76, p. 7]) ln Γ ( x ) =
12 ln ( π ) − x + (cid:18) x − (cid:19) ln x + q X k = (− ) k − B k + k ( k + ) x k + O (cid:18) x q + (cid:19) . Remark . A similar asymptotic expansion of the left-hand sum in (67)can be obtained using the general form of GregoryâĂŹs formula (see Propo-sition 6.20). Setting m = in this expansion, we then retrieve the Gregoryformula-based series expression of Σg given in Proposition 6.8. One of the different versions of Wallis’s formula is given by the following limit(see, e.g., [32, p. 21]) lim n → ∞ · · · · ( n − ) · · · · ( n ) √ n = √ π . (68)The following proposition gives an analogue of this formula in the additivenotation for any function g lying in ∪ p > ( C ∩ D p ∩ K p ) . Proposition 8.17.
Let g ∈ C ∩ D p ∩ K p for some p ∈ N . Let ~ g : R + → R be the function defined by the equation ~ g ( x ) = g ( x ) . Let also h : N ∗ → R be the sequence defined by the equation h ( n ) = σ [ ~ g ] − σ [ g ] + Z ( g ( n + t ) − g ( t )) dt + p X j = G j (cid:0) ∆ j − g ( n + ) − ∆ j − g ( n + ) (cid:1) . hen we have lim n → ∞ h ( n ) + n X k = (− ) k − g ( k ) ! = (69) Proof.
It is clear that the function ~ g also lies in C ∩ D p ∩ K p . We then have n X k = (− ) k − g ( k ) = n X k = g ( k ) − n X k = g ( k ) = ( Σg )( n + ) − ( Σ ~ g )( n + ) . Using (65), we then obtain the claimed formula.Formula (69) actually holds for infinitely many sequences n h ( n ) . Indeed,if it holds for a sequence h ( n ) , then it also holds for the sequence h ( n ) + n − q for any q ∈ N ∗ . Thus, to obtain an elegant analogue of Wallis’s formula, itis advisable to choose h among the simplest functions. For instance, we couldconsider the sequence obtained from the series expansion for h ( n ) about infinityafter removing all the summands that vanish at infinity. Example 8.18. If g ( x ) = ln x , then we have h ( n ) = n ln 2 n + n + −
12 ln ( n + ) + ln ( n + ) − +
12 ln ( π )=
12 ln ( πn ) + O (cid:18) n (cid:19) . Replacing h ( n ) with
12 ln ( πn ) in (69) as recommended, we retrieve Wallis’sformula. If g ( x ) = H x is the harmonic number function, then we have h ( n ) = H n + +
12 ln 2 + ln ( n + ) − ψ ( n + )= ( γ + ln n ) + O (cid:18) n (cid:19) . We then obtain the following analogue of Wallis’s formula lim n → ∞ − ln n + n X k = (− ) k H k ! = γ , which provides an alternative definition of Euler’s constant γ . To give an ad-ditional example, if g ( x ) = H ( ) x = ζ ( ) − ζ ( x + ) is the harmonic numberfunction of order , we obtain the following analogue of Wallis’s formula lim n → ∞ n X k = (− ) k H ( ) k = π
224 . .6 Euler’s reflection formula Recall that the identity Γ ( z ) Γ ( − z ) = π s ( πx ) holds for any z ∈ C \ Z . Thisidentity, known as Euler’s reflection formula (see, e.g., [76, p. 3]), can be provedfor instance by using the Weierstrassian form of the gamma function.Motivated by this and similar examples, it is then natural to wonder if ananalogue of Euler’s reflection formula holds for any multiple log Γ -type function,at least on the interval (
0, 1 ) . Unfortunately, we do not have any answer to thisinteresting question. Thus, results along this line would be most welcome.Actually, reflection formulas may take various forms. For instance, for thedigamma function ψ we have ψ ( x ) − ψ ( − x ) = − π ot ( πx ) , (70)while for the Barnes G -function, we have ln G ( + x ) − ln G ( − x ) = x ln ( π ) − Z x πt ot ( πt ) dt . (71)We also observe that the right sides of some reflection formulas are -periodic.Now, given a function g in ∪ p > ( D p ∩ K p ) , the discussion above suggestssearching for an expression for either Σg ( x ) ± Σg ( − x ) or Σg ( + x ) ± Σg ( − x ) on the interval (
0, 1 ) by means of the Eulerian form or the Weierstrassian form of Σg . If the resulting expression is rather simple, then we have found a reflectionformula for Σg on (
0, 1 ) and we may try to find an extension of this formula to amore general domain by analytic continuation. For instance, using the Eulerianform of the digamma function ψ ( x ) = − γ − x + ∞ X k = (cid:18) k − x + k (cid:19) , we obtain ψ ( x ) − ψ ( − x ) = − x + − x + ∞ X k = (cid:18) − x + k + − x + k (cid:19) and we can show (see, e.g., [15, p. 4] and [34, Eq. (6.88)]) that the latter expres-sion reduces to the -periodic function − π ot ( πx ) , thus retrieving the reflectionformula (70) on the interval (
0, 1 ) , which can then be extended to the domain C \ Z .Regarding reflection formulas involving -periodic functions, we can makethe following interesting observation. Let f , g : C \ Z → C be two complexfunctions and suppose that ∆f = g on C \ Z . Define the functions h + , h − : C \ Z → C by h ± ( z ) = f ( z ) ± f ( − z ) . Then we have ∆ z h ± ( z ) = g ( z ) ∓ g (− z ) andhence the function h + (resp. h − ) is -periodic if and only if g is even (resp.odd). 72 .7 Gauss’ digamma theorem The following formula, due to Gauss, enables one to compute the values of thedigamma function ψ for rational arguments. If a , b ∈ N ∗ with a < b , then wehave ψ (cid:16) ab (cid:17) = − γ − ln ( b ) − π aπb + ⌊ ( b − ) / ⌋ X j = (cid:16) jπ ab (cid:17) ln (cid:18) sin jπb (cid:19) (72)(see, e.g., [45, p. 95] and [76, p. 30]). This formula can be extended to allintegers a , b ∈ N ∗ by means of the difference equation ψ ( x + ) − ψ ( x ) = x .For instance, we have ψ (cid:18) (cid:19) = − γ + π − It is natural to wonder if an analogue of formula (72) holds for any multiple log Γ -type function. Finding an analogue as beautiful as this formula seems tobe hard. However, we have the following partial result. Proposition 8.19.
Let g ∈ D ∩ K and let a , b ∈ N ∗ with a < b . Then Σg (cid:16) ab (cid:17) = b b − X j = (cid:16) − ω − ajb (cid:17) S bj [ g ] , where ω b = e πib and S bj [ g ] = ∞ X k = ω jkb g (cid:18) kb (cid:19) . Proof.
By definition of the map Σ , we have Σg (cid:16) ab (cid:17) = lim n → ∞ n − X k = g (cid:18) bkb (cid:19) − n − X k = g (cid:18) bk + ab (cid:19)! = lim n → ∞ bn − X k = ( u b ( k ) − u b ( k − a )) g (cid:18) kb (cid:19) , where u b ( k ) = , if b divides k , and u b ( k ) = , otherwise; that is, u b ( k ) = b b − X j = ω jkb . This completes the proof.Proposition 8.19 provides a first step in the search for an explicit expressionfor Σg ( ab ) . Depending upon the function g , more computations may be nec-essary to obtain a useful expression. In this respect, the derivation of formula(72) by means of Proposition 8.19 can be found in [57, p. 13].73 xample 8.20. Let us apply Proposition 8.19 to the function g s ( x ) = − x − s ,where s > . This function lies in D ∩ K and we have Σg s ( x ) = ζ ( s , x ) − ζ ( s ) ;see Example 1.3. Let a , b ∈ N ∗ with a < b . For j =
0, . . . , b − , we then have S bj [ g s ] = − b s Li s ( ω jb ) , where Li s ( z ) is the polylogarithm function. Using Proposition 8.19, we thenobtain ζ (cid:16) s , ab (cid:17) = ζ ( s ) − b s − b − X j = (cid:16) − ω − ajb (cid:17) Li s ( ω jb )= b s − b − X j = ω − ajb Li s ( ω jb ) . The inverse conversion formula is simply given by Li s ( ω jb ) = b − s b X k = ω jkb ζ (cid:18) s , kb (cid:19) , j =
1, . . . , b − In the framework of Γ -type functions, Webster [80, Section 8] investigated themultiplicative version of the functional equation f ( x ) + f ( x + ) = h ( x ) , x > and, more generally, of the functional equation m − X j = f (cid:18) x + jm (cid:19) = h ( x ) , x > (73)for any m ∈ N ∗ , where h is a given function satisfying certain conditions. Onthis subject, we present the following result, a variant of which was establishedby Webster [80, Theorem 8.1] in the case when p = . Theorem 8.21 (Webster’s functional equation) . Let p ∈ N , m ∈ N ∗ , and h ∈ ∪ q > ( D q ∩ K q ) be such that ∆h ∈ D p ∩ K p + (resp. ∆h ∈ D p ∩ K p − ). Thenthere is a unique solution to equation (73) lying in K p , namely f ( x ) = ( Σh ) (cid:18) x + m (cid:19) − ( Σh )( x ) . Moreover, this solution lies in K p − (resp. K p + ). roof. Let g : R + → R and g m : R + → R be defined by the equations g ( x ) = h (cid:18) x + m (cid:19) − h ( x ) and g m ( x ) = g (cid:16) xm (cid:17) , respectively. It is easy to see that g m lies in D p ∩ K p + (resp. D p ∩ K p − ) and henceso does g . Let f : R + → R be a solution to equation (73). Then necessarily g ( x ) = m − X j = ∆ j f (cid:18) x + jm (cid:19) = ∆f ( x ) . If f lies in K p , then by the uniqueness and existence theorems and Proposi-tion 5.5, there exists c ∈ R such that f ( x ) = c + ( Σh ) (cid:18) x + m (cid:19) − ( Σh )( x ) . (74)But the function f specified by (74) satisfies (73) if and only if c = .Theorem 8.21 can be somewhat generalized by considering the functionalequation m − X j = f ( x + aj ) = h ( x ) , x > for some a > . Indeed, if we define the function g : R + → R by the equation g ( x ) = h ( amx + a ) − h ( amx ) , we see that any solution f : R + → R to theequation above satisfies g ( x ) = ∆ x f ( amx ) . It then remains to add appropriateassumptions on function h to ensure the uniqueness of the solution. Example 8.22.
We can show that the unique convex or decreasing solution tothe functional equation f ( x ) f ( x + a ) x p = x > a > p > is the function f ( x ) = Γ ( x a ) √ a Γ ( x + a a ) ! p . This result was established by Thielman [78] (see also [4]). The special casewhen p = was previously shown by Mayer [59]. We now apply our results to certain multiple Γ -type functions and multiple log Γ -type functions that are known to be well-studied special functions, namely: thegamma function, the digamma function, the polygamma functions, the Barnes75 -function, the Hurwitz zeta function and its higher derivatives, the generalizedStieltjes constants, and the Catalan number function. We also introduce andinvestigate the principal indefinite sum of the Hurwitz zeta function. For recentbackground on some of these functions, see, e.g., Srivastava and Choi [76].Further examples will be briefly discussed in Section 10.All these examples illustrate how powerful are some of our results to produceformulas and identities methodically. Although many of these formulas andidentities are already known, they had never been derived from such a generaland unified setting.We begin this section with gathering our most relevant and useful results toperform a systematic treatment of these special functions. log Γ -type functions Let g ∈ C r ∩ D p ∩ K p for some p , r ∈ N . Based on the results of this paper, wecan now describe the steps to follow in order to investigate certain propertiesof the function Σg . Note that the function g can also be chosen from a givenmultiple log Γ -type function F by taking g = ∆F . ID card.
Given a function g ∈ C r ∩ D p ∩ K p for some p , r ∈ N , we determinethe asymptotic degree of g and, whenever possible, a simple expression for Σg . Characterization.
A characterization of the function Σg as a solution to the dif-ference equation ∆f = g immediately follows from the uniqueness Theorem 3.1.This characterization states that if f : R + → R is a solution to the equation ∆f = g , then it lies in K p if and only if f = c + Σg for some c ∈ R . Asymptotic constant, generalized Stirling’s and Euler’s constants, Raabe’sformula.
Expressions for σ [ g ] , σ [ g ] , and γ [ g ] can be obtained from Eqs. (36),(65), (66), Definition 6.14, and Propositions 6.23 and 6.25. Recall also that γ [ g ] is subject to inequality (50) and that σ [ g ] is defined if and only if g is integrableat . An integral form of γ [ g ] is given in Proposition 6.24. Finally, the analogueof Raabe’s formula is identity (37). Restriction to the natural integers.
The restriction of Σg to N ∗ is given in(24). Series representations are given in Propositions 6.8 and 6.18. If deg g = ,we also have the representation given in (51). Derivatives of Σg ( x ) at x = . A formula for the derivatives of Σg ( x ) at x = is given in Proposition 7.4. If Σg ( x ) is real analytic at x = , then wecan also write the Taylor series expansion of Σg ( x + ) about x = . Also, theexponential generating function for the sequence n σ [ g ( n ) ] is given in (58). Asymptotic analysis.
The asymptotic behavior of Σg is summarized in Theo-rems 6.2 and 6.5 and Proposition 6.25. The Binet-like function J p + [ Σg ]( x ) isgiven in (38). As shown in Corollary 7.6, the convergence formulas stated in76heorems 6.2 and 6.5 can also be differentiated to derive the asymptotic behav-ior of the derivatives of Σg . In particular, the Binet-like function J p + [ Σg ]( x ) and its derivatives vanish at infinity. Further asymptotic results, including theanalogue of Burnside’s formula, are given in Corollary 6.13 and Proposition 6.26. Eulerian and Weierstrassian forms (series and infinite product representa-tions).
The Eulerian and Weierstrassian forms are given in Theorems 8.1, 8.3,and 8.5. These series can be integrated and differentiated term by term. Also,the analogue of Gauss’ limit for the gamma function is given by the definitionof Σg as the limit of the sequence n f pn [ g ] . Alternative series expression and Fontana-Mascheroni’s series.
These se-ries representations are given in Proposition 6.8.
Alternative representation.
An alternative expression (e.g., an integral rep-resentation) for Σg can sometimes be obtained from Theorem 7.8 and Corol-lary 7.10 by first searching for an appropriate solution to the equation ∆ϕ = g ( r ) . Gauss’ multiplication formula.
A general multiplication formula is given inboth Theorem 8.9 and its companion Proposition 8.12. It should be noted,however, that this formula leads to an interesting identity only when a rathersimple expression for Σg m , where g m ( x ) = g ( xm ) , is available. In addition, anasymptotic expansion of Σg is given in Proposition 8.14. Wallis’s and reflection formulas.
These formulas are discussed in Subsec-tions 8.5 and 8.6.
Webster’s functional equation.
This part is described in Theorem 8.21.
As the gamma function was Webster’s motivating example in his introduction ofthe Γ -type functions, it is natural to test our results on this function first. Notethat Webster also mentioned the q -gamma functions as noteworthy examplesof Γ -type functions. Recall that for any < q < the q -gamma function Γ q isdefined by the equation ln Γ q ( x ) = Σ x − q x − q , x > The following investigation of the gamma function does not reveal quite newformulas. However, it clearly demonstrates how our results can be used to carryout this investigation in a systematic way.
ID card. g ( x ) Membership deg g Σg ( x ) ln x C ∞ ∩ D ∩ K ∞ Γ ( x ) haracterization. A characterization of the gamma function is given in Bohr-Mollerup-Artin’s theorem (see Example 3.2).
Asymptotic constant, generalized Stirling’s and Euler’s constants, Raabe’sformula. exp ( σ [ g ]) σ [ g ] γ [ g ] √ π − +
12 ln ( π ) γ [ g ] = σ [ g ] We have the inequality | σ [ g ] |
12 ln 2 and the following representations σ [ g ] = lim n → ∞ (cid:0) ln n ! + n − − (cid:0) n + (cid:1) ln n (cid:1) ; σ [ g ] = P ∞ k = (cid:0) − (cid:0) k + (cid:1) ln (cid:0) + k (cid:1)(cid:1) ; σ [ g ] = R ∞ (cid:0)
12 ln ( ⌊ t ⌋ + ⌊ t ⌋ ) − ln t (cid:1) dt ; σ [ g ] = R ∞ t − ⌊ t ⌋ − / t dt ; σ [ g ] = R
10 ln Γ ( t + ) dt . Also, Raabe’s formula is given in (64).
Restriction to the natural integers.
For any n ∈ N we have Γ ( n + ) = n ! .Gregory’s formula states that for any n ∈ N ∗ and any q ∈ N we have ln n ! = − n + ( n + ) ln n − q X j = G j (cid:0) ∆ j − g ( n ) − ∆ j − g ( ) (cid:1) − R q , n , with | R q , n | G q | ∆ q g ( n ) − ∆ q g ( ) | . Moreover, Proposition 6.8 gives the following series representation ln n ! =
12 ln ( π ) − n + ( n + ) ln n − ∞ X k = G k + ∆ k g ( n ) , n ∈ N ∗ . (75)Finally, recall Liu’s formula (see Subsection 6.4) ln n ! =
12 ln ( π ) + (cid:18) n + (cid:19) ln n − n + Z ∞ n − { t } t dt . Derivatives of Σg ( x ) at x = . We have ψ ( ) = ( ln Γ ) ′ ( ) = − σ [ g ′ ] = − γ and,for any integer k > , ψ k − ( ) = ( ln Γ ) ( k ) ( ) = (− ) k ( k − ) ! − σ [ g ( k ) ]= (− ) k ( k − ) ! ζ ( k ) , and hence σ [ g ( k ) ] = (− ) k ( k − ) ! ( − ( k − ) ζ ( k )) . ln Γ ( x + ) about x = is ln Γ ( x + ) = − γx + ∞ X k = ζ ( k ) k (− x ) k , | x | < Integrating this equation on (
0, 1 ) , we obtain ∞ X k = (− ) k k ( k + ) ζ ( k ) = γ − +
12 ln ( π ) . Also, the exponential generating function for the sequence n σ [ g ( n ) ] is egf σ [ g ]( x ) = σ [ g ] − x + ( x + ) ln ( x + ) − ln Γ ( x + ) . Asymptotic analysis.
For every a > , we have Γ ( x + a ) ∼ x a Γ ( x ) ∼ √ π e − x x x + a − as x → ∞ ;ln Γ ( x + a ) ∼ x ln x − x as x → ∞ . We also have the results given in Examples 6.3 and 6.7.Considering Binet’s function J ( x ) = ln Γ ( x ) −
12 ln ( π ) + x − (cid:0) x − (cid:1) ln x , for any x > we also have the inequalities (cid:12)(cid:12) ln Γ (cid:0) x + (cid:1) −
12 ln ( π ) + x − x ln x (cid:12)(cid:12) | J ( x ) | (cid:12)(cid:12) ln (cid:0) + x (cid:1)(cid:12)(cid:12) , which confirm that Burnside’s formula (43) provides a better approximation of ln Γ ( x ) than Stirling’s formula.Since all the derivatives of J ( x ) vanish at infinity, for any k ∈ N ∗ we get ψ ( x ) − ln x → and ψ k ( x ) → as x → ∞ . Eulerian and Weierstrassian forms.
For any x > , we have Γ ( x ) = x ∞ Y k = ( + k ) x + xk = e − γx x ∞ Y k = e xk + xk and the corresponding series can be integrated and differentiated term by term(see Examples 8.2 and 8.6). These identities can also be written as follows Γ ( x ) = lim n → ∞ n ! n x x ( x + ) · · · ( x + n ) = lim n → ∞ n ! e xψ ( n ) x ( x + ) · · · ( x + n ) . (cid:18) + n (cid:19) − ⌈ x ⌉ | x − | Γ ( x ) ( n − ) ! n x x ( x + ) ··· ( x + n − ) (cid:18) + n (cid:19) ⌈ x ⌉ | x − | . Alternative series expression and Fontana-Mascheroni’s series.
Identity(75) is also valid for a real argument: for any x > we have ln Γ ( x ) =
12 ln ( π ) − x + x ln x − ∞ X n = G n + ∆ n g ( x )=
12 ln ( π ) − x + x ln x − ∞ X n = | G n + | n X k = (− ) k (cid:0) nk (cid:1) ln ( x + k ) (see Example 6.9). Setting x = in this identity yields the analogue of Fontana-Mascheroni series: ∞ X n = | G n + | n X k = (− ) k (cid:0) nk (cid:1) ln ( k + ) = − +
12 ln ( π ) . Alternative representation.
Considering the antiderivative of the solution ϕ = ψ to the equation ∆ϕ = g ′ , we obtain ln Γ ( x ) = ψ − ( x ) = Z x ψ ( t ) dt . Gauss’ multiplication formula.
As described in Example 8.13, for any m ∈ N ∗ and any x > , we have m − Y j = Γ (cid:18) x + jm (cid:19) = ( π ) m − m − x Γ ( x ) . Also, Corollary 8.10 provides the following formula for any x > Γ ( mx ) m ∼ e − x x x m x as m → N ∞ , which also follows from Stirling’s formula. Moreover, Proposition 8.14 yieldsthe following asymptotic expansion as x → ∞ . For any m , q ∈ N ∗ we have m m − X j = Γ (cid:18) x + jm (cid:19) =
12 ln ( π ) + x ln x − x − m ln x + q X k = (− ) k − B k + k ( k + ) x k m k + + O (cid:18) x q + (cid:19) . m = in this formula, we obtain (see, e.g., [76, p. 7]) ln Γ ( x ) =
12 ln ( π ) − x + (cid:18) x − (cid:19) ln x + q X k = (− ) k − B k + k ( k + ) x k + O (cid:18) x q + (cid:19) . Thus, we have ln Γ ( x ) =
12 ln ( π ) − x + (cid:18) x − (cid:19) ln x + x − x + O (cid:18) x (cid:19) , which is consistent with the analogue of Stirling’s formula − log Γ ( x ) +
12 ln ( π ) − x + (cid:18) x − (cid:19) ln x → as x → ∞ . Wallis’s product formula.
The original Wallis formula is presented in (68).
Reflection formula.
For any x ∈ (
0, 1 ) , we have Γ ( x ) Γ ( − x ) = π s ( πx ) . Webster’s functional equation.
For any m ∈ N ∗ , there is a unique solution f : R + → R + to the equation Q m − j = f ( x + jm ) = x such that ln f is eventuallymonotone, namely f ( x ) = Γ ( x + m ) Γ ( x ) . More generally, for any m ∈ N ∗ and any a > , there is a unique solution f : R + → R + to the equation Q m − j = f ( x + aj ) = x such that ln f is eventuallymonotone, namely f ( x ) = ( am ) m Γ ( xam + m ) Γ ( xam ) . Let us now see what we get if we apply our results to the digamma function x ψ ( x ) and the harmonic number function x H x . Recall first that theidentity H x − = ψ ( x ) + γ holds for any x > . ID card. g ( x ) Membership deg g Σg ( x ) /x C ∞ ∩ D ∩ K ∞ − H x − = ψ ( x ) + γ Characterization.
The digamma function can be characterized as follows:
All eventually monotone solutions f : R + → R to the equation f ( x + ) − f ( x ) = /x are of the form f ( x ) = c + ψ ( x ) , where c ∈ R . x > , H x − = ψ ( x ) + γ = Z
10 1 − t x − − t dt . Indeed, each of the three expressions above vanishes at x = and is an eventuallyincreasing solution to the equation f ( x + ) − f ( x ) = /x . Hence, they mustcoincide on R + . We can prove the following Gauss representation (see, e.g., [76,p. 26]) similarly ψ ( x ) = Z ∞ (cid:18) e − t t − e − xt − e − (cid:19) dt , x > Kairies [44] obtained a variant of the characterization of the digamma functionabove by replacing the eventual monotonicity with the convexity property. Thisvariant is also immediate from our results since g also lies in D ∩ K . Asymptotic constant, generalized Stirling’s and Euler’s constants, Raabe’sformula. σ [ g ] σ [ g ] γ [ g ] ∞ γ γ We have the following representations γ = lim n → ∞ n X k = k − ln n ! = ∞ X k = (cid:18) k − ln (cid:18) + k (cid:19)(cid:19) ; γ = Z ∞ (cid:18) ⌊ t ⌋ − t (cid:19) dt = Z H t dt . Also, the analogue of Raabe’s formula is Z x + x ψ ( t ) dt = ln x , x > We also have for any q ∈ N and any x > J q + [ Σg ]( x ) = ψ ( x ) − ln x + q X j = | G j | B ( x , j ) , where ( x , y ) B ( x , y ) is the beta function. Restriction to the natural integers.
For any n ∈ N we have H n = P nk = k .Gregory’s formula states that for any n ∈ N ∗ and any q ∈ N we have H n − = ln n − q X j = | G j | (cid:18) B ( n , j ) − j (cid:19) − R q , n , | R q , n | G q (cid:12)(cid:12)(cid:12)(cid:12) B ( n , q + ) − q (cid:12)(cid:12)(cid:12)(cid:12) . Derivatives of Σg ( x ) at x = . We have ψ ( ) = − γ and, for any k ∈ N ∗ , ψ k ( ) = (− ) k − ( k − ) ! − σ [ g ( k ) ] = (− ) k − k ! ζ ( k + ) and hence σ [ g ( k ) ] = (− ) k − ( k − ) ! ( − k ζ ( k + )) . The Taylor series expansion of ψ ( x + ) about x = is H x = ψ ( x + ) + γ = ∞ X k = (− ) k − ζ ( k + ) x k , | x | < Integrating this equation on (
0, 1 ) , we retrieve Euler’s series representation of γγ = ∞ X k = (− ) k ζ ( k ) k . Also, the exponential generating function for the sequence n σ [ g ( n ) ] is egf σ [ g ]( x ) = ln ( x + ) − ψ ( x + ) . Asymptotic analysis.
For any a > and any x > , we have | ψ ( x + a ) − ψ ( x ) | ⌈ a ⌉ x and | ψ ( x ) − ln x | x . Considering the value p = in Theorem 6.5, we see that the latter inequalitycan be refined into − x + x ( x + ) ψ ( x ) − ln x − ( x + ) . We also have ψ ( x + a ) − ψ ( x ) → , ψ ( x ) − ln x → , and ψ ( x + a ) ∼ ln x as x → ∞ . Since all the derivatives of J [ Σg ] vanish at infinity, so do the functions ψ k for any k ∈ N ∗ . Finally, for any x > we also have the inequalities (cid:12)(cid:12) ψ (cid:0) x + (cid:1) − ln x (cid:12)(cid:12) | J [ Σg ]( x ) | x , which shows that the analogue of Burnside’s formula ψ ( x ) − ln ( x − ) → as x → ∞ , provides a better approximation of ψ than generalized Stirling’s formula.83 ulerian and Weierstrassian forms. For any x > , we have ψ ( x ) = − γ − x + ∞ X k = (cid:18) k − x + k (cid:19) = − x + ∞ X k = (cid:18) ln (cid:18) + k (cid:19) − x + k (cid:19) and these series can be integrated and differentiated term by term. In particular,we retrieve the product form of e ψ ( x ) obtained in Example 8.2. Also, theinequality in Theorem 3.4 gives (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ( x ) + γ + x − n − X k = (cid:18) k − x + k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ⌈ x ⌉ n , x > n ∈ N ∗ . Alternative series expression and Fontana-Mascheroni’s series.
Proposi-tion 6.8 yields the following series representation ψ ( x ) = ln x − ∞ X n = | G n | B ( x , n ) = ln x − ∞ X n = | G n | n (cid:0) x + n − n (cid:1) , x > Setting x = in this identity, we retrieve Fontana-Mascheroni series: γ = ∞ X n = | G n | n . Setting x = , we get − ln 2 = ∞ X n = | G n | n + which is consistent with the last identity given in Example 6.11. Alternative representation.
We have H x − = H x − x = ψ ( x ) + γ . Gauss’ multiplication formula.
For any m ∈ N ∗ and any x > , we have (see,e.g., [15, p. 5]) m − X j = ψ (cid:18) x + jm (cid:19) = m ( ψ ( x ) − ln m ) and m − X j = H ( x + j ) /m = m ( H x + m − − ln m ) . Also, Corollary 8.10 provides the following formula for any x > m → ∞ ( H mx − − H m − ) = ln x . x → ∞ .For any m , q ∈ N ∗ we have m m − X j = ψ (cid:18) x + jm (cid:19) = ln x + q X k = (− ) k − B k k ( mx ) k + O (cid:18) x q + (cid:19) . Setting m = in this formula, we obtain (see, e.g., [76, p. 36]) ψ ( x ) = ln x + q X k = (− ) k − B k k x k + O (cid:18) x q + (cid:19) . Wallis’s product formula.
The analogue of Wallis’s formula is the classicalidentity ∞ X k = (− ) k − k = ln 2 . Interestingly, the analogue of Wallis’s formula for the function g ( x ) = ψ ( x ) gives lim n → ∞ − ln ( n ) + n X k = (− ) k ψ ( k ) ! = γ , which provides yet another formula to define Euler’s constant γ . This latterformula is obtained by first considering the duplication formula ψ ( x ) = ψ ( x )+ ψ ( x + ) + . Reflection formula.
For any x ∈ (
0, 1 ) , we have ψ ( x ) − ψ ( − x ) = − π ot ( πx ) . Webster’s functional equation.
For any m ∈ N ∗ , there is a unique eventuallymonotone solution f : R + → R to the equation P m − j = f ( x + jm ) = x , namely f ( x ) = ψ (cid:18) x + m (cid:19) − ψ ( x ) . More generally, for any m ∈ N ∗ and any a > , there is a unique eventuallymonotone solution f : R + → R to the equation P m − j = f ( x + aj ) = x , namely f ( x ) = am ψ (cid:18) xam + m (cid:19) − am ψ (cid:16) xam (cid:17) . Example 9.1.
Suppose we wish to prove that ln Γ ( x ) ∼ x ψ ( x ) − x as x → ∞ .Considering the function g ( x ) = ψ ( x ) + /x , we have deg g = and Σg ( x ) = x ψ ( x ) − x + + γ . Then, the generalized Stirling formula yields ( x ψ ( x ) − x + ) − ln Γ ( x ) − ln x + ψ ( x ) → −
12 ln ( π ) as x → ∞ . ln Γ ( x ) , we obtain the claim asymptotic equivalence. Wecan also derive the equivalence ln Γ ( x ) ∼ ( x − ) ψ ( x ) − x as x → ∞ from taking the derivative of the generalized Stirling formula applied to g ( x ) = ln Γ ( x ) . Finally, we also have the equivalence ln Γ ( x ) ∼ x ln x − x as x → ∞ , which is nothing other than Proposition 6.26 with g ( x ) = ln x . We now investigate the polygamma functions ψ ν ( ν ∈ Z ). Our results willprove to be particularly useful when ν < − since, in this case, the function ψ ν has a strictly positive asymptotic degree.For any ν ∈ Z , we set g ν = ∆ψ ν ; hence g ′ ν = g ν + and ψ ′ ν = ψ ν + . Wethen have Σg ν ( x ) = ψ ν ( x )− ψ ν ( ) . (The cases ν = and ν = − correspond tothe functions ψ and ln Γ , respectively, and have been already considered above.)Let us deal with the cases ν ∈ N ∗ and ν ∈ Z \ N separately. In the latter case,we often consider the value ν = − for simplicity and brevity. ν ∈ N ∗ ID card. g ν ( x ) Membership deg g ν Σg ν ( x )(− ) ν ν ! x − ν − C ∞ ∩ e D − N ∩ K ∞ − ψ ν ( x ) − ψ ν ( ) Recall that ψ ν ( ) = (− ) ν + ν ! ζ ( ν + ) (cf. derivatives of ψ ( x ) at x = ). Characterization.
The function ψ ν can be characterized as follows: All eventually monotone solutions f : R + → R to the equation f ( x + ) − f ( x ) = g ν ( x ) are of the form f ( x ) = c ν + ψ ν ( x ) , where c ν ∈ R . This characterization enables us to prove almost immediately the following iden-tity ψ ν ( x ) = (− ) ν − Z ∞ t ν e − xt − e − dt , x > Asymptotic constant, generalized Stirling’s and Euler’s constants, Raabe’sformula. σ [ g ν ] σ [ g ν ] γ [ g ν ] ∞ g ν − ( ) − ψ ν ( ) γ [ g ν ] = σ [ g ν ]
86e have the inequality | σ [ g ν ] | ν ! and the following representations σ [ g ν ] = lim n → ∞ ( P nk = g ν ( k ) + g ν − ( )) ; σ [ g ν ] = P ∞ k = ( g ν ( k ) + g ν − ( k ) − g ν − ( k + )) ; σ [ g ν ] = (− ) ν ν ! R ∞ (cid:0) ⌊ t ⌋ − ν − − t − ν − (cid:1) dt ; σ [ g ν ] = R ( ψ ν ( t + ) − ψ ν ( )) dt . Also, the analogue of Raabe’s formula is Z x + x ψ ν ( t ) dt = g ν − ( x ) , x > We also have for any q ∈ N and any x > J q + [ Σg ν ]( x ) = ψ ν ( x ) − g ν − ( x ) + q X j = G j ∆ j − g ν ( x ) . Restriction to the natural integers.
For any n ∈ N ∗ , we have ψ ν ( n ) − ψ ν ( ) = (− ) ν ν ! n − X k = k − ν − Gregory’s formula states that for any n ∈ N ∗ and any q ∈ N we have n − X k = g ν ( k ) = g ν − ( n ) − g ν − ( )− q X j = G j (cid:0) ∆ j − g ν ( n ) − ∆ j − g ν ( ) (cid:1) − R q , n , with | R q , n | G q | ∆ q g ν ( n ) − ∆ q g ν ( ) | . Derivatives of Σg ν ( x ) at x = . We have ψ ν ( ) = (− ) ν + ν ! ζ ( ν + ) and, forany k ∈ N ∗ , ψ ν + k ( ) = g ν + k − ( ) − σ [ g ( k ) ν ] = (− ) ν + k − ( ν + k ) ! ζ ( ν + k + ) and hence σ [ g ( k ) ν ] = g ν + k − ( ) + (− ) ν + k ( ν + k ) ! ζ ( ν + k + ) . The exponential generating function for the sequence n σ [ g ( n ) ν ] is egf σ [ g ν ]( x ) = g ν − ( x + ) − ψ ν ( x + ) . symptotic analysis. For any a > and any x > , we have | ψ ν ( x + a ) − ψ ν ( x ) | ⌈ a ⌉ | g ν ( x ) | and | ψ ν ( x ) − g ν − ( x ) | | g ν ( x ) | . Considering the value p = in Theorem 6.5, we see that the latter inequalitycan be refined into (cid:12)(cid:12)(cid:12)(cid:12) ψ ν ( x ) − g ν − ( x ) + g ν ( x ) (cid:12)(cid:12)(cid:12)(cid:12) | ∆g ν ( x ) | . We also have, ψ ν ( x ) → and ψ ν ( x + a ) ∼ g ν − ( x ) as x → ∞ . Finally, for any x > we have the inequalities (cid:12)(cid:12) ψ ν (cid:0) x + (cid:1) − g ν − ( x ) (cid:12)(cid:12) | J [ Σg ν ]( x ) | | g ν ( x ) | , which shows that the analogue of Burnside’s formula ψ ν ( x ) − g ν − ( x − ) → as x → ∞ , provides a better approximation of ψ ν than generalized Stirling’s formula. Eulerian and Weierstrassian forms.
For any x > , we have ψ ν ( x ) = − ∞ X k = g ν ( x + k ) . and this series can be integrated and differentiated term by term. Alternative series expression and Fontana-Mascheroni’s series.
Proposi-tion 6.8 gives the following series representation: for any x > we have ψ ν ( x ) = g ν − ( x ) − ∞ X n = G n + ∆ n g ν ( x )= g ν − ( x ) − ∞ X n = | G n + | n X k = (− ) k (cid:0) nk (cid:1) g ν ( x + k ) . Setting x = in this identity yields the analogue of Fontana-Mascheroni series.For instance, taking ν = , we derive the identity ∞ X n = | G n | H n n = π − Taking ν = , we obtain ∞ X n = | G n | ψ ( n + ) − H n n = − ζ ( ) + γ π
26 . auss’ multiplication formula. Differentiating the multiplication formula for ψ , we obtain the following formula. For any m ∈ N ∗ and any x > , we have m − X j = ψ ν (cid:18) x + jm (cid:19) = m ν + ψ ν ( x ) . Also, Corollary 8.10 provides the following limit lim m → ∞ m ν ψ ν ( mx ) = g ν − ( x ) , x > Moreover, Proposition 8.14 yields the following asymptotic expansion as x → ∞ .For any m , q ∈ N ∗ we have m m − X j = ψ ν (cid:18) x + jm (cid:19) = q X k = B k m k k ! g ν + k − ( x ) + O ( g ν + q ( x )) . Setting m = in this formula, we obtain ψ ν ( x ) = q X k = B k k ! g ν + k − ( x ) + O ( g ν + q ( x )) . Wallis’s product formula.
We have ∞ X k = (− ) k − g ν ( k ) = (− ) ν ( − − ν ) ν ! ζ ( ν + ) , that is, ∞ X k = (− ) k − g ν ( k ) = (− ) ν ν ! η ( ν + ) , where η is Dirichlet’s eta function. Reflection formula.
Differentiating the reflection formula for ψ , we obtain thefollowing formula. For any x ∈ (
0, 1 ) , we have ψ ν ( x ) − (− ) ν ψ ν ( − x ) = − π D ν ot ( πx ) . Webster’s functional equation.
For any m ∈ N ∗ , there is a unique eventuallymonotone solution f : R + → R to the equation P m − j = f ( x + jm ) = g ν ( x ) , namely f ( x ) = ψ ν (cid:18) x + m (cid:19) − ψ ν ( x ) . .4.2 Case ν ∈ Z \ N ID card. g ν ( x ) Membership deg g ν Σg ν ( x ) see below C ∞ ∩ D − ν ∩ K ∞ − ν − ψ ν ( x ) − ψ ν ( ) Using (37), we obtain the following recursive way to compute g ν . For anyinteger ν − , g ν − ( x ) = Z x + x ψ ν ( t ) dt = Z x g ν ( t ) dt + Z ψ ν ( t ) dt = Z x g ν ( t ) dt + ψ ν − ( ) . Solving this recurrence equation, we obtain g − ( x ) = ln x and for any integer ν − , g ν ( x ) = Z x ( x − t ) − ν − (− ν − ) ! ln t dt + − ν − X j = ψ ν + j ( ) x j j ! . For instance, g − ( x ) = x ln x − x +
12 ln ( π ) and g − ( x ) = x x − x + (cid:18) x + (cid:19) ln ( π ) + ln A . Characterization.
The function ψ ν can be characterized as follows: All solutions f : R + → R to the equation f ( x + ) − f ( x ) = g ν ( x ) that lie in K − ν are of the form f ( x ) = c + ψ ν ( x ) , where c ∈ R . Asymptotic constant, generalized Stirling’s and Euler’s constants, Raabe’sformula. σ [ g ν ] σ [ g ν ] γ [ g ν ] ψ ν − ( ) − ψ ν ( ) g ν − ( ) − ψ ν ( ) σ [ g ν ] − P − νj = G j ∆ j − g ν ( ) σ [ g − ] σ [ g − ] γ [ g − ] ln A −
14 ln ( π ) ln A +
14 ln ( π ) −
34 ln A +
16 ln 2 − We have | γ [ g ν ] | G − ν | ∆ − ν g ν ( ) | , γ [ g ν ] = Z ∞ − ν X j = G j ∆ j g ν ( ⌊ t ⌋ ) − g ν ( t ) dt , σ [ g ν ] = Z ( ψ ν ( t + ) − ψ ν ( )) dt , σ [ g ν ] = lim n → ∞ n − X k = g ν ( k ) + g ν − ( ) − g ν − ( n ) + − ν X j = G j ∆ j − g ν ( n ) , σ [ g ν ] = − ν X j = G j ∆ j − g ν ( )+ ∞ X k = − ν X j = G j ∆ j g ν ( k ) + g ν − ( k ) − g ν − ( k + ) . Also, the analogue of Raabe’s formula is Z x + x ψ ν ( t ) dt = g ν − ( x ) , x > We also have for any q ∈ N and any x > J q + [ Σg ν ]( x ) = ψ ν ( x ) − g ν − ( x ) + q X j = G j ∆ j − g ν ( x ) . For instance, J [ Σg − ]( x ) = ψ − ( x ) − ( x + ) ln ( x + ) + ( x − ) − x ( x − ) ln x − x ln ( π ) − ln A . Derivatives of Σg ν ( x ) at x = . For any k ∈ N ∗ we have ψ k − ( ) = ( Σg − ) ( k ) ( ) = g ( k − )− ( ) − σ [ g ( k )− ] . We have σ [ g ′ − ] = σ [ ln ] = − +
12 ln ( π ) , σ [ g ′′ − ] = γ , and for any integer k > , σ [ g ( k )− ] = (− ) k − ( k − ) ! ( − ( k − ) ζ ( k − )) . The exponential generating function for the sequence n σ [ g ( n ) ν ] is egf σ [ g ν ]( x ) = g ν − ( x + ) − ψ ν ( x + ) . Integrating this equation for ν = − on (
0, 1 ) (i.e., we use (60)), we obtain aftersome algebra ∞ X k = (− ) k ζ ( k ) k ( k + )( k + ) = γ − +
14 ln ( π ) + ln A . symptotic analysis. For every a > , we have ψ ν ( x + a ) − ψ ν ( x ) − − ν − X j = (cid:0) aj + (cid:1) ∆ j g ν ( x ) → as x → ∞ ; g ν ( x + a ) − − ν − X j = (cid:0) aj (cid:1) ∆ j g ν ( x ) → as x → ∞ . For instance, when ν = − the first limit reduces to Z x + ax ln Γ ( t ) dt − a ln (cid:18) √ π x x e x (cid:19) − (cid:0) a (cid:1) ln (cid:18) ( x + ) x + e x x (cid:19) → as x → ∞ ,with equality if a ∈ {
0, 1, 2 } . Also, for any x > , we have (cid:12)(cid:12) J − ν + [ Σg ν ]( x ) (cid:12)(cid:12) G − ν (cid:12)(cid:12) ∆ − ν g ν ( x ) (cid:12)(cid:12) and ψ ν ( x ) − g ν − ( x ) + − ν X j = G j ∆ j − g ν ( x ) → as x → ∞ ; ∆g ν − ( x ) − − ν − X j = G j ∆ j g ν ( x ) → as x → ∞ . Also, ψ ν ( x + a ) ∼ g ν − ( x ) as x → ∞ . For instance, if ν = − , then the first limit above reduces to ψ − ( x ) − ( x + ) ln ( x + ) + ( x − ) − x ( x − ) ln x − x ln ( π ) → ln A . Eulerian and Weierstrassian forms.
For any x > , we have ψ ν ( x ) − ψ ν ( ) = − g ν ( x ) + − ν − X j = (cid:0) xj + (cid:1) ∆ j g ν ( )+ ∞ X k = − g ν ( x + k ) + − ν X j = (cid:0) xj (cid:1) ∆ j g ν ( k ) and ψ ν ( x ) − ψ ν ( ) = − g ν ( x ) + − ν − X j = (cid:0) xj + (cid:1) ∆ j g ν ( ) − γ (cid:0) x − ν (cid:1) + ∞ X k = − g ν ( x + k ) + − ν − X j = (cid:0) xj (cid:1) ∆ j g ν ( k ) + (cid:0) x − ν (cid:1) k . ν = − , these identities reduce to ψ − ( x ) = ln ( π ) x ( e )( x ) x x ∞ Y k = ( + /k ) ( k + ) ( x )( + x/k ) x + k ( + /k ) ( k + ) x ( x − ) ! and ψ − ( x ) = ln ( π ) x e − γ ( x ) x x ∞ Y k = e k ( x ) ( + /k ) ( k + ) x ( + x/k ) x + k ! . Integrating both the Eulerian and Weierstrassian forms of ψ − ( x ) = ln Γ ( x ) , weobtain the following representations (which are simpler than the previous onessince less terms are involved; see also Example 8.2) ψ − ( x ) = ln e x x x ∞ Y k = e x ( + /k ) x / ( + x/k ) x + k ! = ln e − γx / e x x x ∞ Y k = e x + x / ( k ) ( + x/k ) x + k ! . We also have the analogue of Gauss’ limit for the gamma function ψ − ( x ) = x − x ln x + lim n → ∞ n − X k = (cid:16) x − ( x + k ) ln (cid:16) + xk (cid:17)(cid:17) + x
22 ln n ! . Alternative series expression and Fontana-Mascheroni’s series.
Here theformulas are the same as in the case when ν ∈ N ∗ . Gauss’ multiplication formula.
For any m ∈ N ∗ and any x > , we have m − X j = ψ ν (cid:18) x + jm (cid:19) = m − X j = ψ ν (cid:18) jm (cid:19) + ψ ν ( ) + Σ x g ν (cid:16) xm (cid:17) . Let us expand this formula in the case when ν = − . First, we have g − (cid:16) xm (cid:17) = m g − ( x ) − x ln mm + m − m ψ − ( ) and hence Σ x g − (cid:16) xm (cid:17) = m ψ − ( x ) − (cid:0) x (cid:1) ln mm + (cid:18) m − m x − (cid:19) ψ − ( ) . Also, we have m − X j = ψ − (cid:18) jm (cid:19) = (cid:18) − m (cid:19) ln A − ln m m + ( m − ) ln (( π ) A ) . ψ − m − X j = ψ − (cid:18) x + jm (cid:19) = (cid:18) − m (cid:19) ln (( π ) x A ) + ( m − ) ln (( π ) A )− m ( x − x + ) ln m + m ψ − ( x ) . This formula can also be derived by integrating the multiplication formula ob-tained from g − ( x ) = ln x . Taking m = , we obtain the following analogue ofLegendre’s duplication formula ψ − (cid:16) x (cid:17) + ψ − (cid:18) x + (cid:19) =
12 ln (( π ) x A ) + ln (( π ) A )− ( x − x + ) ln 2 + ψ − ( x ) . Setting x = in this latter identity, we obtain ψ − (cid:18) (cid:19) =
524 ln 2 +
32 ln A +
14 ln π . We also observe that Proposition 8.14 yields the same asymptotic expansionsas in the case when ν ∈ N ∗ . Wallis’s product formula.
We have for instance lim n → ∞ h ( n ) + n X k = (− ) k − g − ( k ) ! =
112 ln 2 − A . where h ( n ) = (cid:0) n + (cid:1) ln n − n ( − ln 2 ) . Incidentally, the analogue of Wallis’sformula for the function g ( x ) = ψ − ( x ) is lim n → ∞ h ( n ) + n X k = (− ) k − ψ − ( k ) ! = ln A −
112 ln 2 , where h ( n ) = n ( n ) − n + n ln ( π ) −
112 ln n . This latter formula is alittle harder to obtain than the former one; it requires the computation of bothfunctions Σψ − ( x ) and Σ x ψ − ( x ) using Corollary 7.10 with r = . That is, Σψ − ( x ) = − x ( x − )( x − ) + x ( x + ) ln ( π )+ x ln A + ( x − ) ψ − ( x ) − ψ − ( x ) and Σ x ψ − ( x ) = − x ( x − )( x − ) + ( x + ) ln A + (− x + x + ) ln 2 − ψ − ( x )+ x ψ − ( x ) − ψ − (cid:18) x + (cid:19) − ψ − ( x ) . eflection formula. A reflection formula can be obtained by integrating theidentity ln Γ ( x ) + ln Γ ( − x ) = ln π − ln sin ( πx ) . For example, for any x ∈ (
0, 1 ) ,we have ψ − ( x ) − ψ − ( − x ) = x ln π −
12 ln ( π ) − Z x ( πt ) dt . In particular, we obtain R /
20 ln sin ( πt ) dt = −
12 ln 2 . Webster’s functional equation.
For any m ∈ N ∗ , there is a unique solution f : R + → R to the equation P m − j = f ( x + jm ) = g ν ( x ) that lies in K − ν − , namely f ( x ) = ψ ν (cid:18) x + m (cid:19) − ψ ν ( x ) . G -function The Barnes function G : R + → R + is the function G = /Γ (see Subsection 5.2).Hence, it can be defined by the equation ln G = Σ ln Γ = Σψ − . ID card. g ( x ) Membership deg g Σg ( x ) ln Γ ( x ) C ∞ ∩ D ∩ K ∞ G ( x ) Characterization.
The function G can be characterized as follows: All solutions f : R + → R + to the equation f ( x + ) = Γ ( x ) f ( x ) forwhich ln f lies in K are of the form f ( x ) = c G ( x ) , where c > . Asymptotic constant, generalized Stirling’s and Euler’s constants, Raabe’sformula. exp ( σ [ g ]) σ [ g ] γ [ g ] e / ( π ) − / A − +
14 ln ( π ) − A γ [ g ] = σ [ g ] We have the inequality | γ [ g ] |
512 ln 2 as well as the following representations σ [ g ] =
12 ln ( π ) + lim n → ∞ n X k = Γ ( k ) − ψ − ( n ) −
12 ln Γ ( n ) −
112 ln n ! ; σ [ g ] = ∞ X k = Γ ( k ) e k √ k (cid:0) + k (cid:1) k k √ π ; σ [ g ] = Z ∞ (cid:18) ln Γ ( ⌊ t ⌋ ) Γ ( t ) + ln ⌊ t ⌋ / ⌊ t + ⌋ / (cid:19) dt ; σ [ g ] = Z
10 ln G ( t + ) dt . Z x + x ln G ( t ) dt = σ [ g ] + ψ − ( x ) , x > We also have for any q ∈ N and any x > J q + [ Σg ]( x ) = ln G ( x ) − ψ − ( x ) +
14 ln ( π ) − + A + q X j = G j ∆ j − g ( x ) . For instance, J [ Σg ]( x ) = ln G ( x ) − ψ − ( x ) +
14 ln ( π ) − + A +
12 ln Γ ( x ) −
112 ln x . Note that the functions ln G ( x ) and ψ − ( x ) are strongly related (see (76) below)in the sense that we can easily express one of it in terms of the other. Restriction to the natural integers.
For any n ∈ N ∗ we have G ( n ) = Q n − k = k ! . Derivatives of Σg ( x ) at x = . For any k ∈ N ∗ we have ( Σg ) ( k ) ( ) = g ( k − ) ( ) − σ [ g ( k ) ] . We also have σ [ g ′ ] = σ [ ψ ] = ( − ln ( π )) , σ [ g ′′ ] = σ [ ψ ] = and for anyinteger k > , σ [ g ( k ) ] = (− ) k ( k − )( k − ) ! ζ ( k − ) . The Taylor series expansion of ln G ( x + ) about x = is (see, e.g., [76, p. 311]) ln G ( x + ) = ( ln ( π ) − ) x − γ + x − ∞ X k = ζ ( k ) k + (− x ) k + | x | < Integrating this equation on (
0, 1 ) , we obtain ∞ X k = (− ) k ζ ( k )( k + )( k + ) = + γ − A . Also, the exponential generating functions for the sequences n σ [ g ( n ) ] and n γ [ g ( n ) ] are egf σ [ g ]( x ) = ln G ( x + ) − ψ − ( x + ) +
14 ln ( π ) − + A and egf γ [ g ]( x ) = egf σ [ g ]( x ) + γ x , (
0, 1 ) (i.e., we use (60)),we obtain after some algebra ∞ X k = (− ) k k − k ( k + )( k + ) ζ ( k ) = − A −
14 ln ( π ) . Asymptotic analysis.
For any x > and any a > we have (cid:18) + x (cid:19) − ⌈ a ⌉ | ( a − ) | G ( x + a ) G ( x ) Γ ( x ) a x ( a ) (cid:18) + x (cid:19) ⌈ a ⌉ | ( a − ) | , with equality if a ∈ {
1, 2 } . Thus, G ( x + a ) ∼ G ( x ) Γ ( x ) a x ( a ) as x → ∞ , In view of Wendel’s inequalities for the gamma function (see Example 6.3),we conjecture that the inequalities above can be simplified and tightened byreplacing the extreme functions by ( + a/x ) − | ( a − ) | and ( + a/x ) | ( a − ) | .We also have | J [ Σg ]( x ) |
512 ln (cid:0) + x (cid:1) , x > that is, in the multiplicative notation, (cid:18) + x (cid:19) − / G ( x ) Γ ( x ) / e ψ − ( x )+ σ [ g ] x / (cid:18) + x (cid:19) /
12 , x > Thus, we obtain the following analogues of Stirling’s formula G ( x ) ∼ exp ( ψ − ( x ) + σ [ g ]) Γ ( x ) − x as x → ∞ ; G ( x + ) ∼ exp ( ψ − ( x ) + σ [ g ]) Γ ( x ) x as x → ∞ . Using the definition of G in terms of ψ − ( x ) (see (76) below) as well as theStirling formula for the gamma function, we obtain the following simpler form G ( x + ) ∼ A − x x − ( π ) x e − x + as x → ∞ . We also have, for any a > G ( x + a ) ∼ ψ − ( x ) as x → ∞ . Finally, recall that all the derivatives of J [ Σg ]( x ) vanish at infinity. For instance,the first derivative yields the convergence result ln Γ ( x ) − (cid:18) x − (cid:19) ψ ( x ) + x → ( + ln ( π )) as x → ∞ x ψ ( x ) → as x → ∞ . Eulerian and Weierstrassian forms.
For any x > , we have G ( x ) = Γ ( x ) ∞ Y k = Γ ( k ) Γ ( x + k ) k x ( + /k )( x )= e (− γ − ) ( x ) Γ ( x ) ∞ Y k = Γ ( k ) Γ ( x + k ) k x e ψ ( k ) ( x ) . Also, the analogue of Gauss’ limit for the gamma function is G ( x ) = lim n → ∞ Γ ( ) Γ ( ) · · · Γ ( n ) Γ ( x ) Γ ( x + ) · · · Γ ( x + n ) n ! x n ( x ) . Alternative series expression and Fontana-Mascheroni’s series.
Using Propo-sition 6.8, we also derive the following product representation: for any x > we have ln G ( x ) = ψ − ( x ) + σ [ g ] −
12 ln Γ ( x ) − ∞ X n = G n + ∆ n + g ( x )= ψ − ( x ) + σ [ g ] −
12 ln Γ ( x ) − ∞ X n = | G n + | n X k = (− ) k (cid:0) nk (cid:1) ln ( x + k ) . In the multiplicative notation: G ( x ) = exp ( ψ − ( x ) + σ [ g ]) Γ ( x ) − x (cid:18) x + x (cid:19) − × (cid:18) ( x + ) x ( x + ) (cid:19) (cid:18) ( x + )( x + ) ( x + ) x (cid:19) − · · · Setting x = in this identity yields the analogue of Fontana-Mascheroni series: σ [ g ] = −
12 ln ( π ) + ∞ X n = | G n + | n X k = (− ) k (cid:0) nk (cid:1) ln ( k + ) . Alternative representation.
Considering the antiderivative of the solution ϕ ( x ) = ( x − ) ψ ( x ) − x + +
12 ln ( π ) to the equation ∆ϕ = g ′ = ψ , we obtain ln G ( x ) = − (cid:0) x (cid:1) + ( x − ) ln Γ ( x ) +
12 ln ( π ) x − ψ − ( x ) . (76)This identity can also be proved directly using the characterization result; in-deed, both sides vanish at x = and are eventually -convex solutions to theequation f ( x + ) − f ( x ) = ln Γ ( x ) . Hence, they must coincide on R + .98 auss’ multiplication formula. For any m ∈ N ∗ and any x > , we have m − Y j = G (cid:18) x + jm (cid:19) = e Σ x ln Γ ( xm ) m − Y j = G (cid:18) jm (cid:19) . For instance, setting m = in this identity, we obtain ln G (cid:18) x + (cid:19) + ln G (cid:16) x (cid:17) − ln G (cid:18) (cid:19) = Σ x ln Γ (cid:16) x (cid:17) . However, to make this multiplication formula interesting and usable, we need tofind a simple expression for its right side. In particular, we need a closed-formexpression for the function Σ x ln Γ ( xm ) .Proposition 8.14 yields the following asymptotic expansion as x → ∞ . Forany m , q ∈ N ∗ we have m m − X j = G (cid:18) x + jm (cid:19) = σ [ g ] + q X k = B k m k k ! ψ k − ( x ) + O ( ψ q − ( x )) . Setting m = in this formula, we obtain ln G ( x ) = σ [ g ] + q X k = B k k ! ψ k − ( x ) + O ( ψ q − ( x )) . Thus, we have ln G ( x ) = σ [ g ] + ψ − ( x ) − ψ − ( x ) + ψ ( x ) − ψ ( x ) + O (cid:18) x (cid:19) , which is consistent with the analogue of Stirling’s formula − ln G ( x ) + σ [ g ] + ψ − ( x ) − ψ − ( x ) +
112 ln ( x ) → as x → ∞ . Wallis’s product formula.
Using Legendre’s duplication formula for the gammafunction, we obtain Σ x ln Γ ( x ) = ln G ( x ) + ln G ( x + ) − ln G ( )+ ( x + ) ln 2 − x ( π ) . Using this identity, we can derive the surprising analogue of Wallis’s formula lim n → ∞ Γ ( ) Γ ( ) · · · Γ ( n − ) Γ ( ) Γ ( ) · · · Γ ( n ) (cid:18) ne (cid:19) n = √ Incidentally, the analogue of Wallis’s formula for the function g ( x ) = ln G ( x ) is lim n → ∞ G ( ) G ( ) · · · G ( n − ) G ( ) G ( ) · · · G ( n ) n n − n −
124 2 n − π n e n − n − = A
12 , Σ ln G ( x ) and Σ x ln G ( x ) usingCorollary 7.10 with r = . That is, Σ ln G ( x ) = − x ( x − )( x − ) + x ( x − ) ln ( π ) − x ln A + ( x − )( x − ) ln Γ ( x ) − ( x − ) ψ − ( x ) + ψ − ( x ) and Σ x ln G ( x ) = − x ( x − )( x − ) − x ln A + ( x − x − ) ln 2 + x ( x − ) ln π +
12 ln Γ ( x ) + ( x − )( x − ) ln Γ ( x )− ( x − ) ψ − ( x ) + ψ − ( x ) . Reflection formula.
A reflection formula for the Barnes G -function is given in(71); see, e.g., [76, p. 45]. Webster’s functional equation.
For any m ∈ N ∗ , there is a unique solution f : R + → R + to the equation Q m − j = f ( x + jm ) = Γ ( x ) such that ln f lies in K ,namely f ( x ) = G ( x + m ) G ( x ) . For any x > , the function s ζ ( s , x ) is defined as an analytic continuationto C \ { } of the series (see, e.g., [76]) ∞ X k = ( x + k ) − s = Γ ( s ) Z ∞ t s − e − xt − e − t dt , ℜ ( s ) > It is known (see, e.g., [76, p. 160]) that this function satisfies the differenceequation ζ ( s , x + ) − ζ ( s , x ) = − x − s , x > (77)For any fixed s ∈ R \ { } , define the function g s : R + → R by g s ( x ) = − x − s .We then have g s ∈ K ∞ . If s > and s = , then g s ∈ D N . If s > , then g s ∈ e D − N . If − p < s < for some p ∈ N , then g s ∈ D p N , and hence we canconsider p = + deg g s = ⌊ − s ⌋ . In all cases, we have Σg s ( x ) = ζ ( s , x ) − ζ ( s ) .100 D card. g s ( x ) Membership deg g s Σg s ( x )− x − s C ∞ ∩ e D − N ∩ K ∞ , if s > C ∞ ∩ D ⌊ − s ⌋ ∩ K ∞ , if s < − + ⌊ − s ⌋ + ζ ( s , x ) − ζ ( s ) Characterization.
The function ζ ( s , x ) can be characterized as follows: All solutions f s : R + → R to the equation f s ( x + ) − f s ( x ) = − x − s that lie in K ⌊ − s ⌋ + are of the form f s ( x ) = c s + ζ ( s , x ) , where c s ∈ R .Asymptotic constant, generalized Stirling’s and Euler’s constants, Raabe’sformula. σ [ g s ] σ [ g s ] γ [ g s ] ∞ , if s > − ζ ( s ) , if s <
1. 1 s − − ζ ( s ) σ [ g s ] − P ⌊ − s ⌋ + j = G j ∆ j − g s ( ) We have the inequality | γ [ g s ] | G ⌊ − s ⌋ + | ∆ ⌊ − s ⌋ + g s ( ) | as well as the following representations σ [ g s ] = Z ( ζ ( s , t + ) − ζ ( s )) dt , γ [ g s ] = Z ∞ (cid:18) ⌊ − s ⌋ + X j = G j ∆ j g s ( ⌊ t ⌋ ) − g s ( t ) (cid:19) dt , σ [ g s ] = lim n → ∞ − n − s s − − n − X k = k − s − ⌊ − s ⌋ + X j = G j ∆ j − g s ( n ) , σ [ g s ] = ⌊ − s ⌋ + X j = G j ∆ j − g s ( )+ ∞ X k = k − s − ( k + ) − s s − + ⌊ − s ⌋ + X j = G j ∆ j g s ( k ) . Also, the analogue of Raabe’s formula is Z x + x ζ ( s , t ) dt = x − s s − x > q ∈ N and any x > J q + [ Σg s ]( x ) = ζ ( s , x ) − x − s s − + q X j = G j ∆ j − g ( x ) . Restriction to the natural integers.
For any n ∈ N ∗ we have ζ ( s , n ) − ζ ( s ) = − n − X k = k − s and ζ ( s , n ) = ∞ X k = n k − s . Gregory’s formula states that for any n ∈ N ∗ and any q ∈ N we have n − X k = k − s = − n − s s − + q X j = G j (cid:0) ∆ j − g s ( n ) − ∆ j − g s ( ) (cid:1) + R qs , n , with | R qs , n | G q | ∆ q g s ( n ) − ∆ q g s ( ) | . Moreover, Proposition 6.8 gives the following series representation n − X k = k − s = ζ ( s ) − x − s s − + ∞ X k = G k ∆ k − g s ( n ) , n ∈ N ∗ . (78) Derivatives of Σg s ( x ) at x = . We have ( Σg s ) ( k ) ( ) = (− s ) k ζ ( s + k ) , k ∈ N ∗ , and σ [ g ( k ) s ] = − (− s ) k − ( + (− s − k + ) ζ ( s + k )) , k ∈ N . The Taylor series expansion of ζ ( s , x + ) about x = is ζ ( s , x + ) = ∞ X k = (cid:0) − sk (cid:1) ζ ( s + k ) x k , | x | < Integrating this equation on (
0, 1 ) , we obtain the identity ∞ X k = (cid:0) − sk (cid:1) ζ ( s + k ) k + = s − s < s / ∈ Z . (When s > , the summand in the series above does not approach zero as k increases.) Also, the exponential generating function for the sequence n σ [ g ( n ) s ] is egf σ [ g s ]( x ) = ( x + ) − s s − − ζ ( s , x + ) . symptotic analysis. For any a > and any x > , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ ( s , x + a ) − ζ ( s , x ) − ⌊ − s ⌋ + X j = (cid:0) aj (cid:1) ∆ j − g s ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ⌈ a ⌉ (cid:12)(cid:12)(cid:12)(cid:0) a − ⌊ − s ⌋ + (cid:1)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∆ ⌊ − s ⌋ + g s ( x ) (cid:12)(cid:12)(cid:12) . In particular, ζ ( s , x + a ) − ζ ( s , x ) − ⌊ − s ⌋ + X j = (cid:0) aj (cid:1) ∆ j − g s ( x ) → as x → ∞ , with equality if a ∈ {
1, 2, . . . , ⌊ − s ⌋ + } . Also, for any x > , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ ( s , x ) − x − s s − + ⌊ − s ⌋ + X j = G j ∆ j − g s ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G ⌊ − s ⌋ + (cid:12)(cid:12)(cid:12) ∆ ⌊ − s ⌋ + g s ( x ) (cid:12)(cid:12)(cid:12) , from which we derive the following analogue of Stirling’s formula ζ ( s , x ) − x − s s − + ⌊ − s ⌋ + X j = G j ∆ j − g s ( x ) → as x → ∞ . In particular, if s > , then ζ ( s , x ) → as x → ∞ .The results above enable us to investigate the asymptotic behavior of ζ ( s , x ) for large values of x . For instance, when s = − we obtain that for every a > the expression ζ (cid:0) −
32 , x + a (cid:1) + x / + (cid:0) a − (cid:0) a (cid:1) − (cid:1) x / + (cid:0)(cid:0) a (cid:1) + (cid:1) ( x + ) / approaches zero as x → ∞ .We also have ζ ( s , x + a ) ∼ x − s s − as x → ∞ . Finally, if s > − , then we have the analogue of Burnside’s formula ζ ( s , x ) − s − ( x − ) − s → as x → ∞ , which provides a better approximation of ζ ( s , x ) than generalized Stirling’s for-mula. Eulerian and Weierstrassian forms. If s > , then for any x > , we simplyhave ζ ( s , x ) = ∞ X k = ( x + k ) − s and this series can be integrated and differentiated term by term. In particular,we observe that ψ ν ( x ) = (− ) ν + ν ! ζ ( ν + x ) , ν ∈ N ∗ , x > s < , then for any x > , we have ζ ( s , x ) − ζ ( s ) = − g s ( x ) + ⌊ − s ⌋ X j = (cid:0) xj + (cid:1) ∆ j g s ( )+ ∞ X k = − g s ( x + k ) + ⌊ − s ⌋ X j = (cid:0) xj (cid:1) ∆ j g s ( k ) and the Weierstrassian form can be obtained similarly. Again, both series canbe integrated and differentiated term by term.For instance, we have ζ (cid:0) −
32 , x (cid:1) − ζ (cid:0) − (cid:1) = x + lim n → ∞ n − X k = (cid:16) ( x + k ) − k (cid:17) − x n − (cid:0) x (cid:1) ∆ n n ! = x − x − ( √ − ) (cid:0) x (cid:1) + ∞ X k = (cid:16) ( x + k ) − k − x∆ k k − (cid:0) x (cid:1) ∆ k k (cid:17) = x − x + ζ (cid:0) (cid:1)(cid:0) x (cid:1) + ∞ X k = (cid:16) ( x + k ) − k − x∆ k k − (cid:0) x (cid:1) k − (cid:17) . Alternative series expression and Fontana-Mascheroni’s series.
Identity(78) is also valid for a real argument: for any x > we have ζ ( s , x ) = x − s s − − ∞ X n = G n + ∆ n g s ( x )= x − s s − + ∞ X n = | G n + | n X k = (− ) k (cid:0) nk (cid:1) ( x + k ) − s . Setting x = in this identity yields a known series expression for ζ ( s ) that isthe analogue of Fontana-Mascheroni series ζ ( s ) = s − + ∞ X n = | G n + | n X k = (− ) k (cid:0) nk (cid:1) ( k + ) − s . Gauss’ multiplication formula.
For any m ∈ N ∗ and any x > , we have m − X j = ζ (cid:18) s , x + jm (cid:19) = m − X j = ζ (cid:18) s , jm (cid:19) + ζ ( s ) + m s ( ζ ( s , x ) − ζ ( s )) . Since P mj = ζ ( s , j/m ) = m s ζ ( s ) , the formula above actually reduces to m − X j = ζ (cid:18) s , x + jm (cid:19) = m s ζ ( s , x ) . lim m → ∞ m s − ζ ( s , mx ) = x − s s − x > s < m → ∞ m s − ( ζ ( s , mx ) − ζ ( s , m )) = x − s − s − x > s = Moreover, Proposition 8.14 yields the following asymptotic expansion as x → ∞ .For any m , q ∈ N ∗ we have m m − X j = ζ (cid:18) s , x + jm (cid:19) = x − s s − − q X k = (cid:0) − sk − (cid:1) B k k m k x k + s − + O (cid:18) x q + s (cid:19) = s − q X k = (cid:0) − sk (cid:1) B k m k x k + s − + O (cid:18) x q + s (cid:19) . Setting m = in this formula, we obtain ζ ( s , x ) = s − q X k = (cid:0) − sk (cid:1) B k x k + s − + O (cid:18) x q + s (cid:19) . Wallis’s product formula. If s > , then we have ∞ X k = (− ) k − k s = ( − − s ) ζ ( s ) = η ( s ) , (79)where s η ( s ) is Dirichlet’s eta function. When s < , the form of the formulastrongly depends upon the value of s . When s = − for instance, we obtain lim n → ∞ h ( n ) + n X k = (− ) k k ! = ( √ − ) ζ (− ) . where h ( n ) = − n + p n . Reflection formula. If s is an integer, then the extension to the domain R \ Z of the function ζ ( s , x ) + (− ) s ζ ( s , 1 − x ) is -periodic. However, no closed-formexpression for this function seems to be known. Webster’s functional equation.
For any m ∈ N ∗ , there is a unique solution f s : R + → R to the equation P m − j = f s ( x + jm ) = − x − s that lies in K ⌊ − s ⌋ + ,namely f s ( x ) = ζ (cid:18) s , x + m (cid:19) − ζ ( s , x ) . Example 9.2.
Consider the function g ( x ) = x ( x + ) − . We then have g ( x ) =( x + ) − ( x + ) + ( x + ) − and hence Σg ( x ) = c − ζ (−
32 , x + ) + ζ (−
12 , x + ) − ζ (
12 , x + ) for some c ∈ R . 105 .7 The generalized Stieltjes constants Recall that the generalized Stieltjes constants are the numbers γ n ( x ) thatoccur in the Laurent series expansion of the Hurwitz zeta function ζ ( s , x ) = s − + ∞ X n = (− ) n n ! γ n ( x )( s − ) n . (80)Here we naturally restrict the values of x to the set R + . Recall also that the num-bers γ n = γ n ( ) , where n ∈ N , are called the Stieltjes constants . For recentbackground on these constants, see, e.g., Blagouchine [16, 17] and Blagouchineand Coppo [19] (see also Nan-Yue and Williams [68]).These constants are known to satisfy γ ( x ) = − ψ ( x ) and γ = γ as well asthe following identities for every q ∈ N γ q = lim n → ∞ n X k = ( ln k ) q k − ( ln n ) q + q + ! , γ q ( x ) = lim n → ∞ n X k = ( ln ( x + k )) q x + k − ( ln ( x + n )) q + q + ! . Interestingly, the generalized Stieltjes constants also satisfy the difference equa-tion γ q ( x + ) − γ q ( x ) = g q ( x ) , where g q : R + → R is the function defined by the equation g q ( x ) = − x ( ln x ) q . Thus, our theory is particularly suitable to investigate these constants. For any q ∈ N , the function g q lies in C ∞ ∩ D ∩ K ∞ and is eventually increasing. Byuniqueness of Σg q , it follows that Σg q ( x ) = γ q ( x ) − γ q . ID card. g q ( x ) Membership deg g q Σg q ( x )− x ( ln x ) q C ∞ ∩ D ∩ K ∞ − γ q ( x ) − γ q Characterization.
The function γ q can be characterized as follows: All eventually monotone solutions f : R + → R to the equation f ( x + ) − f ( x ) = g q ( x ) are of the form f ( x ) = c q + γ q ( x ) , where c q ∈ R . Asymptotic constant, generalized Stirling’s and Euler’s constants, Raabe’sformula. σ [ g q ] σ [ g q ] γ [ g q ] ∞ − γ q − γ q σ [ g q ] is exactly the opposite ofthe Stieltjes constant γ q . We also have the following representations γ q = ∞ X k = (cid:18) ( ln k ) q k − ( ln ( k + )) q + − ( ln ( k )) q + q + (cid:19) ; γ q = Z ∞ (cid:18) ( ln ⌊ t ⌋ ) q ⌊ t ⌋ − ( ln t ) q t (cid:19) dt . The analogue of Raabe’s formula is Z x + x γ q ( t ) dt = − ( ln x ) q + q + x > We also have for any r ∈ N and any x > J r + [ Σg q ]( x ) = γ q ( x ) + ( ln x ) q + q + + r X j = G j ∆ j − g q ( x ) . Derivatives of Σg q ( x ) at x = . We have γ ( k ) q ( ) = g ( k − ) q ( ) − σ [ g ( k ) q ] = − ∞ X j = g ( k ) q ( j ) , k ∈ N ∗ . The exponential generating function for the sequence n σ [ g ( n ) q ] is egf σ [ g q ]( x ) = − γ q ( x + ) − q + ( ln ( x + )) q + Asymptotic analysis.
Let x > be so that g q is increasing on [ x , ∞ ) . Thenfor any a > , we have | γ q ( x + a ) − γ q ( x ) | ⌈ a ⌉ (cid:12)(cid:12)(cid:12)(cid:12) ( ln x ) q x (cid:12)(cid:12)(cid:12)(cid:12) ; (cid:12)(cid:12)(cid:12)(cid:12) γ q ( x ) + ( ln x ) q + q + (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ( ln x ) q x (cid:12)(cid:12)(cid:12)(cid:12) . In particular, we have γ q ( x + a ) − γ q ( x ) → and γ q ( x ) + ( ln x ) q + q + → as x → ∞ . The latter convergence result is the analogue of Stirling’s formula. It expressesthe fact that the function J [ Σg q ] vanishes at infinity. We also note that so doall its derivatives. For instance, we have γ ′ q ( x ) + ( ln x ) q x → as x → ∞ . a > , we have γ q ( x + a ) ∼ − ( ln x ) q + q + as x → ∞ . Finally, for any x > we have the inequalities (cid:12)(cid:12) γ q (cid:0) x + (cid:1) − g q ( x ) (cid:12)(cid:12) | J [ Σg q ]( x ) | | g q ( x ) | , which shows that the analogue of Burnside’s formula γ q ( x ) + q + ( ln ( x − )) q + → as x → ∞ , provides a better approximation of γ q ( x ) for large values of x than the gener-alized Stirling formula. For < x , we use the following approximations(see [68, p. 148]) (cid:12)(cid:12)(cid:12)(cid:12) γ ( x ) − x (cid:12)(cid:12)(cid:12)(cid:12) γ and (cid:12)(cid:12)(cid:12)(cid:12) γ q ( x ) − ( ln x ) q x (cid:12)(cid:12)(cid:12)(cid:12) ( + (− ) q )( q ) ! q q + ( π ) q , n ∈ N ∗ . Eulerian and Weierstrassian forms.
For any x > , we have γ q ( x ) = γ q + ( ln x ) q x + ∞ X k = (cid:18) ( ln ( x + k )) q x + k − ( ln k ) q k (cid:19) and γ q ( x ) = ( ln x ) q x + ∞ X k = (cid:18) ( ln ( x + k )) q x + k − ( ln ( k + )) q + − ( ln k ) q + q + (cid:19) . Both series can be differentiated and integrated term by term. Also, if n ∈ N ∗ is so that g q is increasing on [ n , ∞ ) , then for any x > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ q ( x ) − γ q − ( ln x ) q x − n − X k = (cid:18) ( ln ( x + k )) q x + k − ( ln k ) q k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ⌈ x ⌉ (cid:12)(cid:12)(cid:12)(cid:12) ( ln n ) q n (cid:12)(cid:12)(cid:12)(cid:12) . Alternative series expression and Fontana-Mascheroni’s series.
For any x > satisfying the assumptions of Proposition 6.8, we obtain γ q ( x ) + ( ln x ) q + q + = ∞ X n = G n + ∆ nx ( ln x ) q x = ∞ X n = | G n + | n X k = (− ) k (cid:0) nk (cid:1) ( ln ( x + k )) q x + k . x = in this identity (provided that x = satisfies the assumptionsof Proposition 6.8), we obtain the Fontana-MascheroniâĂŹs series expression of γ q γ q = ∞ X n = | G n + | n X k = (− ) k (cid:0) nk (cid:1) ( ln ( k + )) q k + This latter expression was obtained by Blagouchine [17, p. 383].
Antiderivative of γ q ( x ) . All eventually concave solutions f : R + → R to theequation f ( x + ) − f ( x ) = G q ( x ) , where G q ( x ) = Z x g q ( t ) dt = − ( ln x ) q + q + = Z x + x γ q ( t ) dt are of the form f ( x ) = c q + Z x γ q ( t ) dt for some c q ∈ R . Gauss’ multiplication formula.
The following analogue of Gauss’ multiplica-tion formula was previously known (see also [16, p. 542]) but it can be derivedstraightforwardly from our results. For any m ∈ N ∗ and any x > , we have m − X j = γ q (cid:18) x + jm (cid:19) = − mq + (cid:18) ln 1 m (cid:19) q + + m q X j = (cid:0) qj (cid:1) (cid:18) ln 1 m (cid:19) j γ q − j ( x ) . In particular, m X j = γ q (cid:18) jm (cid:19) = − mq + (cid:18) ln 1 m (cid:19) q + + m q X j = (cid:0) qj (cid:1) (cid:18) ln 1 m (cid:19) j γ q − j . Also, Corollary 8.10 provides the following limits for x > m → ∞ q X j = (cid:0) qj (cid:1) (cid:18) ln 1 m (cid:19) j ( γ q − j ( mx ) − γ q − j ( m )) = − ( ln x ) q + q + m → ∞ − q + (cid:18) ln 1 m (cid:19) q + + q X j = (cid:0) qj (cid:1) (cid:18) ln 1 m (cid:19) j γ q − j ( mx ) = − ( ln x ) q + q + For instance, setting q = in these formulas yields lim m → ∞ γ ( mx ) − γ ( m ) + ( ln m )( ψ ( mx ) − ψ ( m )) = − ( ln x ) m → ∞ γ ( mx ) − ( ln m ) + ψ ( mx ) ln m = − ( ln x ) m = in the multiplication formula, we obtain the following analogueof Legendre’s duplication formula γ q (cid:16) x (cid:17) + γ q (cid:18) x + (cid:19) = − q + (cid:18) ln 12 (cid:19) q + + q X j = (cid:0) qj (cid:1) (cid:18) ln 12 (cid:19) j γ q − j ( x ) . When q = and q = , the multiplication formula reduces to the knownformulas m − X j = ψ (cid:18) x + jm (cid:19) = m ( ψ ( x ) − ln m ) ; m − X j = γ (cid:18) x + jm (cid:19) = − m ( ln m ) + m ( ln m ) ψ ( x ) + m γ ( x ) . Moreover, Proposition 8.14 yields the following asymptotic expansion as x → ∞ .For any m , q ∈ N ∗ we have m m − X j = γ (cid:18) x + jm (cid:19) = ( ln x ) − ( ln x ) m m − X j = ψ (cid:18) x + jm (cid:19) + q X k = (− ) k − B k H k − k ( mx ) k + O (cid:18) x q + (cid:19) . Setting m = in this latter formula, we obtain γ ( x ) = ( ln x ) − ψ ( x ) ln x + q X k = (− ) k − B k H k − k x k + O (cid:18) x q + (cid:19) . Thus, we have γ ( x ) = ( ln x ) − ψ ( x ) ln x − x + x + O (cid:18) x (cid:19) . Wallis’s product formula.
The analogue of Wallis’s formula for the function g q ( x ) is ∞ X k = (− ) k ( ln k ) q k = − ( ln 2 ) q + q + + q − X j = (cid:0) qj (cid:1) ( ln 2 ) q − j γ j . (81)This formula was established by Briggs and Chowla [22, Eq. (8)]. For q = , itreduces to ∞ X k = (− ) k ln kk = − ( ln 2 ) + γ ln 2 . q = , we obtain ∞ X k = (− ) k ( ln k ) k = − ( ln 2 ) + γ ( ln 2 ) + γ These latter two formulas were also established by Hardy [40].As an aside, let us establish conversion formulas between the sequences q γ q and q η ( q ) ( ) , where η ( s ) is the Dirichlet eta function introducedin (79) and η ( q ) ( ) stands for the limiting value of η ( q ) ( s ) as s → . To easethe computations, let us instead consider the conversion formulas between thesequences q γ q and q λ q , where λ q = q + ( ln 2 ) q + + (− ) q + η ( q ) ( ) , q ∈ N . Using (81), we can readily derive the following equations λ q = q − X k = (cid:0) qk (cid:1) ( ln 2 ) q − k γ k , q ∈ N . (82)These equations actually consist of an infinite consistent triangular system.Solving this system provides the following conversion formula γ q = q X k = (cid:0) qk (cid:1) B q − k k + ( ln 2 ) q − k − λ k + q ∈ N , (83)that is, γ q = − B q + q + ( ln 2 ) q + + q X k = (− ) k (cid:0) qk (cid:1) B q − k k + ( ln 2 ) q − k − η ( k + ) ( ) , q ∈ N . Indeed, plugging (83) in the right side of (82) we obtain for any q ∈ N q − X k = (cid:0) qk (cid:1) ( ln 2 ) q − k γ k = q − X k = (cid:0) qk (cid:1) ( ln 2 ) q − k k X j = (cid:0) kj (cid:1) B k − j j + ( ln 2 ) k − j − λ j + = q − X j = (cid:0) qj (cid:1) ( ln 2 ) q − j − λ j + j + q − X k = j (cid:0) q − jk − j (cid:1) B k − j , where the inner sum reduces to q − j − . The latter quantity then reduces to λ q ,as expected. Remark . The conversion formulas (82) and (83) are not new. In essence,they were established by Liang and Todd [54, Eq. (3.6)] and Nan-Yue andWilliams [68, Eqs. (1.9) and (7.1)]. 111 ebster’s functional equation.
For any m ∈ N ∗ , there is a unique eventuallymonotone solution f : R + → R to the equation P m − j = f ( x + jm ) = g q ( x ) , namely f ( x ) = γ q (cid:18) x + m (cid:19) − γ q ( x ) . Rational arguments theorem.
Let us apply Proposition 8.19 to the function g q ( x ) . For any a , b ∈ N ∗ with a < b and any j ∈ {
0, . . . , b − } we have S bj [ g q ] = b (− ) q + q X i = (cid:0) qi (cid:1) ( ln b ) q − i D is Li s ( z ) (cid:12)(cid:12) ( s , z )=( ω jb ) and hence γ q (cid:16) ab (cid:17) − γ q = (− ) q + q X i = (cid:0) qi (cid:1) ( ln b ) q − i b − X j = ( − ω − ajb ) D is Li s ( z ) (cid:12)(cid:12) ( s , z )=( ω jb ) . We note that a more a practical formula was derived in the special case when q = by Blagouchine [16] as a generalization of Gauss’ digamma theorem. Let s ∈ R \ { } and q ∈ N . Differentiating q times both sides of (77) we obtain ζ ( q ) ( s , x + ) − ζ ( q ) ( s , x ) = (− ) q + x − s ( ln x ) q , x > where ζ ( q ) ( s , x ) stands for D qs ζ ( s , x ) . This equation shows that the investiga-tion of the higher derivatives of the Hurwitz zeta function can be carried outusing our results. For an earlier investigation, see, e.g., [15, p. 36 et seq. ]. ID card. g s , q ( x ) Membership deg g s , q Σg s , q ( x )− x − s (− ln x ) q C ∞ ∩ e D − N ∩ K ∞ , if s > C ∞ ∩ D ⌊ − s ⌋ ∩ K ∞ , if s < − + ⌊ − s ⌋ + ζ ( q ) ( s , x )− ζ ( q ) ( s ) We also observe that this investigation can be regarded as a simultaneous gener-alization of the studies of the Hurwitz zeta function and the generalized Stieltjesconstants. For the latter, we observe that (− ) q lim s → g s , q ( x ) = − x ( ln x ) q . Setting s = in our results may also be very informative as it produces formulasinvolving the well-studied quantities ζ ( q ) ( ) and ζ ( q ) ( x ) − ζ ( q ) ( ) for any q ∈ N . Characterization.
The function ζ ( q ) ( s , x ) can be characterized as follows:112 ll solutions f s , q : R + → R to the equation f s , q ( x + ) − f s , q ( x ) = g s , q ( x ) that lie in K ⌊ − s ⌋ + are of the form f s , q ( x ) = c s , q + ζ ( q ) ( s , x ) , where c s , q ∈ R .Asymptotic constant, generalized Stirling’s and Euler’s constants, Raabe’sformula. σ [ g s , q ] σ [ g s , q ] γ [ g s , q ] ∞ , if s > − ζ ( q ) ( s ) , if s < − q ! ( − s ) q + − ζ ( q ) ( s ) σ [ g s , q ] − P ⌊ − s ⌋ + j = G j ∆ j − g s , q ( ) We have σ [ g s , q ] = lim n → ∞ n − X k = g s , q ( k ) − Z n g s , q ( t ) dt + ⌊ − s ⌋ + X j = G j ∆ j − g s , q ( n ) = ⌊ − s ⌋ + X j = G j ∆ j − g s , q ( )− ∞ X k = Z k + k g s , q ( t ) dt − ⌊ − s ⌋ + X j = G j ∆ j g s , q ( k ) . Setting s = in the previous formulas, we obtain (− ) q ( q ! + ζ ( q ) ( )) = lim n → ∞ n X k = ( ln k ) q − Z n ( ln t ) q dt − ( ln n ) q ! = ∞ X k = (cid:18) ( ln k ) q − Z k + k ( ln t ) q dt (cid:19) . On differentiating both sides of (80), we also obtain the following surprisingidentity (− ) q ( q ! + ζ ( q ) ( )) = ∞ X n = γ n + q n ! . We also have Z x g s , q ( t ) dt = q ! − Γ ( q + ( s − ) ln x )( − s ) q + x > and hence the analogue of Raabe’s formula is Z x + x ζ ( q ) ( s , t ) dt = − Γ ( q + ( s − ) ln x )( − s ) q + = − q ! x − s ( − s ) q + q X j = (( s − ) ln x ) j j ! , x > r ∈ N and any x > J r + [ Σg s , q ]( x ) = ζ ( q ) ( s , x ) − Z x + x ζ ( q ) ( s , t ) dt + r X j = G j ∆ j − g s , q ( x ) . Restriction to the natural integers.
For any n ∈ N ∗ we have ζ ( q ) ( s , n ) − ζ ( q ) ( s ) = n − X k = g s , q ( k ) . Gregory’s formula states that for any n ∈ N ∗ and any r ∈ N we have n − X k = g s , q ( k ) = Z n g s , q ( t ) dt − r X j = G j (cid:0) ∆ j − g s ( n ) − ∆ j − g s ( ) (cid:1) − R rs , q , n , with | R rs , q , n | G r | ∆ r g s , q ( n ) − ∆ r g s , q ( ) | . Asymptotic analysis.
We have ζ ( q ) ( s , x + a ) − ζ ( q ) ( s , x ) − ⌊ − s ⌋ + X j = (cid:0) aj (cid:1) ∆ j − g s , q ( x ) → as x → ∞ , with equality if a ∈ {
1, 2, . . . , ⌊ − s ⌋ + } . Also, we have the following analogue ofStirling’s formula ζ ( q ) ( s , x ) − Z x + x ζ ( q ) ( s , t ) dt + ⌊ − s ⌋ + X j = G j ∆ j − g s , q ( x ) → as x → ∞ . Setting s = in this latter formula, we obtain ζ ( q ) ( x ) + Γ ( q + − ln x ) + (− ) q + ( ln x ) q → as x → ∞ . We also have ζ ( q ) ( s , x + a ) ∼ Z x + x ζ ( q ) ( s , t ) dt as x → ∞ . Finally, if s > − , then we have the analogue of Burnside’s formula ζ ( q ) ( s , x ) − R x + x − ζ ( q ) ( s , t ) dt → as x → ∞ , which provides a better approximation of ζ ( s , x ) than the analogue of Stirling’sformula. 114 ulerian and Weierstrassian forms. If s > , then for any x > , we simplyhave ζ ( q ) ( s , x ) = − ∞ X k = g s , q ( x + k ) and this series can be integrated and differentiated term by term. If s < , thenfor any x > , the analogue of Gauss’ limit is ζ ( q ) ( s , x ) − ζ ( q ) ( s ) = − g s , q ( x )+ lim n → ∞ n − X k = ( g s , q ( k ) − g s , q ( x + k )) + ⌊ − s ⌋ X j = (cid:0) xj (cid:1) ∆ j − g s , q ( n ) . Also, the analogue of Euler’s product form is ζ ( q ) ( s , x ) − ζ ( q ) ( s ) = − g s , q ( x ) + ⌊ − s ⌋ X j = (cid:0) xj + (cid:1) ∆ j g s , q ( )+ ∞ X k = − g s , q ( x + k ) + ⌊ − s ⌋ X j = (cid:0) xj (cid:1) ∆ j g s , q ( k ) and the Weierstrassian form can be obtained similarly. Again, the series can beintegrated and differentiated term by term. Note that the case where ( s , q ) =(
0, 2 ) can be found in Ramanujan’s second notebook [15, p. 26–27]. Alternative series expression and Fontana-Mascheroni’s series.
For any x > satisfying the assumptions of Proposition 6.8, we obtain ζ ( q ) ( s , x ) = Z x + x ζ ( q ) ( s , t ) dt − ∞ X n = G n + ∆ n g s , q ( x )= Z x + x ζ ( q ) ( s , t ) dt − ∞ X n = | G n + | n X k = (− ) k (cid:0) nk (cid:1) g s , q ( x + k ) . Setting x = in this identity (provided that x = satisfies the assumptionsof Proposition 6.8) yields a series expression for ζ ( q ) ( s ) that is the analogue ofFontana-Mascheroni series ζ ( q ) ( s ) = − q ! ( − s ) q + − ∞ X n = | G n + | n X k = (− ) k (cid:0) nk (cid:1) g s , q ( k + ) , which can also be obtained differentiating the analogue of Fontana-Mascheroniseries for the Hurwitz zeta function. For instance, we have ζ ′′ ( ) = − + ∞ X n = | G n + | n X k = (− ) k (cid:0) nk (cid:1) ( ln ( k + )) γ − π − ( ln ( π )) + γ Gauss’ multiplication formula.
Upon differentiating the analogue of Gauss’multiplication formula for the Hurwitz zeta function, we immediately obtainthe following multiplication formula. For any m ∈ N ∗ and any x > , we have m − X j = ζ ( q ) (cid:18) s , x + jm (cid:19) = m s q X j = (cid:0) qj (cid:1) ( ln m ) q − j ζ ( j ) ( s , x ) . Also, Corollary 8.10 provides the following limit for any x > and any s < m → ∞ q X j = (cid:0) qj (cid:1) ( ln m ) q − j ζ ( j ) ( s , mx ) m − s = − Γ ( q + ( s − ) ln x )( − s ) q + Also, for any s = , we have lim m → ∞ q X j = (cid:0) qj (cid:1) ( ln m ) q − j ζ ( j ) ( s , mx ) − ζ ( j ) ( s , m ) m − s = q ! − Γ ( q + ( s − ) ln x )( − s ) q + Wallis’s product formula.
When s < , the form of the analogue of Wallis’sproduct formula strongly depends upon the value of s . If s > , then we have η ( q ) ( s ) = ∞ X k = (− ) k − k s (− ln k ) q = ζ ( q ) ( s ) − − s q X j = (cid:0) qj (cid:1) (cid:18) ln 12 (cid:19) q − j ζ ( j ) ( s ) , where s η ( s ) is Dirichlet’s eta function. Just as we did for the formulas (82)and (83), we can easily establish the following conversion formulas for s > µ q ( s ) = q − X k = (cid:0) qk (cid:1) (cid:18) ln 12 (cid:19) q − k ζ ( k ) ( s ) , q ∈ N , ζ ( q ) ( s ) = q X k = (cid:0) qk (cid:1) B q − k k + (cid:18) ln 12 (cid:19) q − k − µ k + ( s ) , q ∈ N , where µ q ( s ) = s − ( ζ ( q ) ( s ) − η ( q ) ( s )) − ζ ( q ) ( s ) , q ∈ N . Webster’s functional equation.
For any m ∈ N ∗ , there is a unique solution f s , q : R + → R to the equation P m − j = f s , q ( x + jm ) = g s , q ( x ) that lies in K ⌊ − s ⌋ + ,namely f s , q ( x ) = ζ ( q ) (cid:18) s , x + m (cid:19) − ζ ( q ) ( s , x ) . .9 The principal indefinite sum of the Hurwitz zeta func-tion For any s ∈ R \ { } , we define the function ζ ( s , · ) : R + → R by the equation ζ ( s , x ) = Σ x ζ ( s , x ) . Thus defined, this function can be studied through our results. Contrary tothe previous examples, here we introduce a completely new function that hasseemingly no closed form in terms of known elementary functions. Hence wegive it a new symbol and a new name. To keep this investigation simple werestrict ourselves to the case when s > , for which the sequence n ζ ( s , n ) issummable. We then introduce κ ( s ) = ∞ X k = ζ ( s , k ) and we note that Z ∞ ζ ( s , t ) dt = ζ ( s − ) s − ID card. g s ( x ) Membership deg g s Σg s ( x ) ζ ( s , x ) C ∞ ∩ e D − N ∩ K ∞ − ζ ( s , x ) Characterization.
The function ζ ( s , x ) can be characterized as follows: All eventually monotone solutions f s : R + → R to the equation f s ( x + ) − f s ( x ) = ζ ( s , x ) are of the form f s ( x ) = c s + ζ ( s , x ) ,where c s ∈ R .Asymptotic constant, generalized Stirling’s and Euler’s constants, Raabe’sformula. σ [ g s ] σ [ g s ] γ [ g s ] ∞ κ ( s ) − ζ ( s − ) s − γ [ g s ] = σ [ g s ] We have the inequality | σ [ g s ] | ζ ( s ) and the following representations σ [ g s ] = Z ζ ( s , t + ) dt = Z ∞ ( ζ ( s , ⌊ t ⌋ ) − ζ ( s , t )) dt . Also, the analogue of Raabe’s formula is Z x + x ζ ( s , t ) dt = κ ( s ) − ζ ( s − x ) s − x > q ∈ N and any x > J q + [ Σg s ]( x ) = ζ ( s , x ) − κ ( s ) + ζ ( s − x ) s − + q X j = G j ∆ j − x ζ ( s , x ) . Derivatives of Σg s ( x ) at x = . We have ( Σg s ) ( k ) ( ) = (− ) k − k ! (cid:0) sk (cid:1) κ ( s + k ) , k ∈ N ∗ , and σ [ g ( k ) s ] = (− ) k − ( k − ) ! (cid:0) sk − (cid:1) ζ ( s + k − ) + (− ) k k ! (cid:0) sk (cid:1) κ ( s + k ) , k ∈ N ∗ . The Taylor series expansion of ζ ( s , x + ) about x = is ζ ( s , x + ) = − ∞ X k = (cid:0) sk (cid:1) κ ( s + k )(− x ) k . Asymptotic analysis.
For any a > and any x > , we have | ζ ( s , x + a ) − ζ ( s , x ) | ⌈ a ⌉ ζ ( s , x ) ; (cid:12)(cid:12)(cid:12)(cid:12) ζ ( s , x ) − κ ( s ) + ζ ( s − x ) s − (cid:12)(cid:12)(cid:12)(cid:12) ζ ( s , x ) . In particular, we have ζ ( s , x ) → κ ( s ) as x → ∞ .We also have ζ ( s , x ) ∼ κ ( s ) − ζ ( s − x − ) s − as x → ∞ . Eulerian and Weierstrassian forms.
For any x > , we have ζ ( s , x ) = κ ( s ) − ∞ X k = ζ ( s , x + k ) . and this series can be integrated and differentiated term by term. In particular, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ ( s − x ) s − − ∞ X k = ζ ( s , x + k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ζ ( s , x ) and ∞ X k = ζ ( s , x + k ) → as x → ∞ . lternative series expression and Fontana-Mascheroni’s series. Proposi-tion 6.8 gives the following series representation: for any x > we have ζ ( s , x ) = κ ( s ) − ζ ( s − x ) s − − ∞ X n = G n + ∆ nx ζ ( s , x )= κ ( s ) − ζ ( s − x ) s − − ∞ X n = | G n + | n X k = (− ) k (cid:0) nk (cid:1) ζ ( s , x + k ) . Setting x = in this identity yields the analogue of Fontana-Mascheroni series: ∞ X n = | G n + | n X k = (− ) k (cid:0) nk (cid:1) ζ ( s , k + ) = κ ( s ) − ζ ( s − ) s − Wallis’s product formula.
We have ∞ X k = (− ) k − ζ ( s , k ) = ( − − s ) ζ ( s ) + ( − − s ) κ ( s )− − s ∞ X k = ζ (cid:18) s , k + (cid:19) . This formula is obtained by combining Proposition 6.4 with the duplicationformula for the Hurwitz zeta function ζ ( s , 2 x ) = − s ζ ( s , x ) + − s ζ (cid:18) s , x + (cid:19) . Webster’s functional equation.
For any m ∈ N ∗ , there is a unique eventuallymonotone solution f : R + → R to the equation P m − j = f ( x + jm ) = ζ ( s , x ) , namely f ( x ) = ζ (cid:18) s , x + m (cid:19) − ζ ( s , x ) . The Catalan number function is the restriction to R + of the map x C x defined on (−
12 , ∞ ) by C x = x + (cid:0) xx (cid:1) . This function satisfies the equation C x + = (cid:18) − x + (cid:19) C x . The additive version of this equation reads ∆f = g , where the function g isthe logarithm of a rational function. We observe that such equations have beenthoroughly investigated by Anastassiadis [6, p. 41] (see also [49]).119 D card. g ( x ) Membership deg g Σg ( x ) ln (cid:0) − x + (cid:1) C ∞ ∩ D ∩ K ∞ C x Characterization.
The function C x can be characterized as follows: All solutions f : R + → R + to the equation ( x + ) f ( x + ) = ( x + ) f ( x ) for which ln f lies in K are of the form f ( x ) = c C x , where c > .Asymptotic constant, generalized Stirling’s and Euler’s constants, Raabe’sformula. exp ( σ [ g ]) σ [ g ] γ [ g ] √ π e / (cid:0) + ln 827 π (cid:1) (cid:0) + ln 427 π (cid:1) We have the inequality | γ [ g ] |
12 ln 54 and the following representations γ [ g ] = R ∞ ( t − ⌊ t ⌋ − ) ( t + )( t + ) dt ; σ [ g ] = R
10 ln C t + dt . Also, Raabe’s formula is Z x + x ln C t dt = ln e ( x + ) x + √ π ( x + ) x + ! , x > Restriction to the natural integers.
For any n ∈ N ∗ we have C n = n + (cid:0) nn (cid:1) . Asymptotic analysis.
For any a > , we have C x + a C x ∼ a and C x ∼ x x / √ π as x → ∞ . Also, the analogue of Burnside’s formula gives ln C x − ln e ( x ) x √ π ( x + ) x + ! → as x → ∞ . Eulerian and Weierstrassian forms.
For any x > , we have C x = x + x + x ∞ Y k = (cid:0) − k + (cid:1) x (cid:0) − k + (cid:1) x − (cid:0) − x + k + (cid:1) and C x = x + x + e − x ∞ Y k = + xk + + x k + e x ( k + )( k + ) . The scope of applications of our theory is very wide since it applies to anyfunction lying in ∪ p > ( D p ∩ K p ) . In Section 9, we have made a thorough studyof some special functions. In the present section, we briefly discuss furtherexamples that the reader may want to explore in more detail. The multiple gamma functions introduced in Subsection 5.2 can also be studiedthrough the sequence of functions G G
1, . . . , defined by (see [76, p. 56]) G p ( x ) = Γ p ( x ) (− ) p − p ∈ N . Equivalently, we have G ( x ) = x and ln G p = Σ ln G p − for all p ∈ N ∗ . Clearly,the function ln G p − lies in C ∞ ∩ D p ∩ K ∞ and we have deg ln G p = p . Also,the sequence can naturally be extended to p = − by setting G − ( x ) = + /x .Just as for the gamma function and the Barnes G -function, we can derivethe following asymptotic equivalence: for any a > , G p ( x + a ) G p ( x ) ∼ p − Y j = G p − j − ( x )( aj + ) as x → ∞ , with equality if a ∈ {
1, 2, . . . , p } . We also have the following product represen-tation G p ( x ) = G p − ( x ) ∞ Y k = G p − ( k ) G p − ( x + k ) G p − ( k ) x G p − ( k )( x ) · · · G − ( k )( xp ) and the recurrence formula ln G p ( x ) = − ( x − ) σ [ D ln G p − ] + Z x ΣD ln G p − ( t ) dt . For example, one can show that ln G ( x ) = − x ( x − )( x − ) + x ( x − ) ln ( π ) + (cid:0) x − (cid:1) ln Γ ( x )− ( x − ) ψ − ( x ) + ψ − ( x ) − x ψ − ( ) . (This formula can also be checked by means of the characterization of G as a -convex solution to the equation ∆f = ln G .) More generally, from the knownexpressions for G , G , G , and G , we can derive the general formula ln G p ( x ) = h p ( x ) + p − X j = (− ) p − − j (cid:0) x − j (cid:1) ( ψ j − p ( x ) − x ψ j − p ( )) , h ( x ) = ln x , h ( x ) = , h ( x ) = − (cid:0) x (cid:1) , and h ( x ) = − x ( x − )( x − ) − x ψ − ( ) + ψ − ( x ) . The hyperfactorial function (or K -function) is the solution K : R + → R to theequation K ( x + ) = x x K ( x ) defined by the identity ln K = Σ∆ ln K . Considerthe function g = ∆ ln K , that is, g ( x ) = x ln x . Using the identity ∆ψ − ( x ) = Z x t dt + ψ − ( ) , we derive g ( x ) = x + ∆ψ − ( x ) − ψ − ( ) , and hence ln K ( x ) = Σg ( x ) = (cid:0) x (cid:1) + ψ − ( x ) − x ψ − ( ) = ( x − ) ln Γ ( x ) − ln G ( x ) . The integer sequence n K ( n ) has entry A002109 in the OEIS database [75]. The Hurwitz Lerch transcendent Φ ( z , s , a ) is a generalization of the Hurwitzzeta function defined as an analytic continuation of the series P ∞ k = z k ( a + k ) − s when | z | < and a ∈ C \ N (see, e.g., [76]). It satisfies the difference equation Φ ( z , s , a + ) − z − Φ ( z , s , a ) = − z − a − s . It follows that the modified function Φ ( z , s , a ) = − z a Φ ( z , s , a ) satisfies thedifference equation Φ ( z , s , a + ) − Φ ( z , s , a ) = z a a − s . Thus, for some real values of z and s , the restriction to R + of the map a Φ ( z , s , a ) fits the assumptions of our theory. Its complete investigation throughour results is left to the reader. Consider the -variable function Q ( x , s ) = Γ ( x , s ) /Γ ( x ) on R + , where Γ ( x , s ) is the upper incomplete gamma function. Thus defined, the function Q ( x , s ) satisfies the difference equation Q ( x + s ) − Q ( x , s ) = e − s s x /Γ ( x + ) . For any s > , we define the function g s : R + → R by g s ( x ) = e − s s x /Γ ( x + ) .This function lies in C ∞ ∩ e D − N ∩ K ∞ and has the property that Σg s ( x ) = Q ( x , s ) − e − s . 122e also note that the Eulerian form of Q ( x , s ) is Q ( x , s ) = − ∞ X k = g s ( x + k ) = − e − s s x Γ ( x + ) ∞ X k = Γ ( x + ) Γ ( x + k + ) s k = − e − s s x Γ ( x + ) ∞ X k = x − k s k . Consider the restriction to (
0, 1 ] × R + of the map ( x , a , b ) I x ( a , b ) = B ( x ; a , b ) B ( a , b ) , where B ( x ; a , b ) is the incomplete beta function. Thus defined, the function I x ( a , b ) satisfies the difference equation I x ( a + b ) − I x ( a , b ) = − x a ( − x ) b a B ( a , b ) . Defining the function g b , x : R + → R by g b , x ( a ) = − x a ( − x ) b / ( a B ( a , b )) forany fixed x and b , we can investigate the difference equation above. We leaveit as an exercise. The function g : R + → R defined by the equation g ( x ) = √ π e − x lies in C ∞ ∩ e D − N ∩ K ∞ and satisfies the equation erf ( x ) = R x g ( t ) dt , where erf is the Gauss error function. For x > we then have Σg ( x ) = √ π ∞ X k = ( e −( k + ) − e −( k + x ) ) The generalized Stirling formula yields the following convergence result erf ( x ) + √ π ∞ X k = e −( k + x ) → as x → ∞ . Also, the analogue of Legendre’s duplication formula provides the surprisingidentity ∞ X k = (− e −( k + x ) + e −( k + ) − e −( k + x + ) + e −( k + ) − e −( k + ) + e −( k + x ) ) = The function g : R + → R defined by the equation g ( x ) = e − x /x lies in C ∞ ∩ e D − N ∩ K ∞ and satisfies the equation E ( x ) = R ∞ x g ( t ) dt , where E is the exponentialintegral. For x > we then have Σg ( x ) = ∞ X k = (cid:18) e −( k + ) k + − e −( k + x ) k + x (cid:19) . The generalized Stirling formula easily provides the following convergence result E ( x ) − ∞ X k = e −( k + x ) k + x → as x → ∞ . Also, the analogue of Raabe’s formula is Z x + x Σg ( t ) dt = − ln ( e − ) − E ( x ) , x > Recall that, for any n ∈ N , the n th degree Bernoulli polynomial B n ( x ) is definedby the equation B n ( x ) = n X k = (cid:0) nk (cid:1) B n − k x k , where B k is the k th Bernoulli number. These polynomials satisfy the differenceequation B n ( x + ) − B n ( x ) = n x n − . Thus, the function g n : R + → R definedby the equation g n ( x ) = n x n − lies in C ∞ ∩ D n ∩ K ∞ and has the propertythat Σg n ( x ) = B n ( x ) − B n ( ) . The form of the function g n also shows that B n ( x ) − B n ( ) = nζ ( − n ) − nζ ( − n , x ) , n ∈ N . Thus, the Bernoulli polynomials can be characterized as follows:
All solutions f n : R + → R to the equation f n ( x + ) − f n ( x ) = n x n − that lie in K n are of the form f n ( x ) = c n + B n ( x ) , where c n ∈ R . We also easily retrieve the multiplication formula: m − X j = B n (cid:18) x + jm (cid:19) = m n − B n ( x ) x > For any n ∈ N , the n th degree Bernoulli polynomial of the second kind isdefined by the equation ψ n ( x ) = Z x + x (cid:0) tn (cid:1) dt . In particular, we have ψ n ( ) = G n . Also, these polynomials satisfy the differ-ence equation ψ n + ( x + ) − ψ n + ( x ) = ψ n ( x ) . Thus, the function g n : R + → R defined by the equation g n ( x ) = ψ n ( x ) lies in C ∞ ∩ D n + ∩ K ∞ and has the property that Σg n ( x ) = ψ n + ( x ) − ψ n + ( ) .Thus, the Bernoulli polynomials of the second kind can be characterized asfollows: All solutions f n : R + → R to the equation f n ( x + )− f n ( x ) = ψ n ( x ) that lie in K n + are of the form f n ( x ) = c n + ψ n + ( x ) , where c n ∈ R .
11 Conclusion
Krull-Webster’s theory has proved to be a very nice and useful contribution tothe resolution of the difference equation ∆f = g on the real half-line R + . In thispaper, we have provided a significant generalization of Krull-Webster’s theory byconsiderably relaxing the asymptotic condition imposed on function g , and wehave demonstrated through various examples how this generalization providesa unified framework to investigate the properties of many special functions.This framework has indeed enabled us to derive several general formulas thatnow constitute a powerful toolbox and even a genuine Swiss Army knife toinvestigate a large variety of functions.The key point of this generalization was the discovery of expression (2) forthe sequences n f pn [ g ]( x ) , p ∈ N . We also observe that our uniquenessand existence results strongly rely on Lemma 2.4 together with identities (13)and (18). These results actually constitute the common core and even thefundamental cornerstone of all the subsequent formulas that we derived in thispaper. For instance, the generalized Stirling formula (40) has been obtainedalmost miraculously by merely integrating both sides of the inequality given inLemma 2.4. Also, Gregory’s summation formula (44) has been derived instantlyby integrating both sides of identity (18), and we have shown how its remaindercan be controlled using Lemma 2.4 again.Our results clearly shed light on the way many of the classical special func-tions, such as the polygamma functions and the derivatives of the Hurwitz zeta125unction, can be systematically studied, sometimes by deriving identities andformulas almost mechanically.Beyond this systematization aspect, our theory has enabled us to introducea number of new important and useful objects. For instance, the map Σ itselfis a new concept that appears to be as fundamental as the basic antiderivativeoperation (cf. Definition 5.1). In this respect, it would be interesting to extendthe map Σ to a larger domain, e.g., a linear space of functions that wouldinclude not only the current admissible functions but also every function thathas an exponential growth. Other concepts such as the Binet-like function andthe asymptotic constant also appear to be new fundamental objects that meritfurther study.In conclusion, we can clearly see that this area of investigation is very in-triguing. We have just skimmed the surface, and there are a lot of questionsthat emerge naturally. We now list a few below. (cid:15) Find necessary and sufficient conditions on function g to ensure both theuniqueness and existence of solutions lying in K p to the equation ∆f = g (cf. Webster’s question in Appendix B). (cid:15) Find general methods to determine analogues of Euler’s reflection formulaand Gauss’ digamma theorem for any multiple log Γ -type function. (cid:15) Find necessary and sufficient conditions on function g for the function Σg to be real analytic. (cid:15) Show how our results can be used and interpreted when extending somemultiple log Γ -type functions to complex domains. Remark . At some places in this paper (e.g., in Theorem 6.5), we havemade the assumption that g (resp. g ( r ) for some r ∈ N ∗ ) is continuous to ensurethe existence of certain integrals. Although we can often relax this conditionby simply requiring that g (resp. g ( r ) ) is locally integrable, we have kept thiscontinuity assumption for simplicity and consistency with similar results wherehigher order differentiability is assumed.Recall also that any monotone function f defined on a compact interval [ a , b ] is integrable. Thus, for any function g lying in ∪ p > ( D p ∩ K p ) , the integral of Σg on [ x , x + ] exists for sufficiently large x . Nevertheless, most of our resultsthat involve this latter integral also use the asymptotic constant σ [ g ] and theintegral of g on the interval [ x ] . Thus, for the sake of simplicity, we havealways ensured integrability on compact intervals by assuming continuity onthe whole of R + . 126 On Krull-Webster’s asymptotic condition
Summary: We show that our uniqueness and existence results fully generalize arecent attempt by Rassias and Trif [72] to solve the particular case when p = . Recall that the asymptotic condition imposed by Krull and Webster onfunction g is that, for each x > , g ( t + x ) − g ( t ) → as t → ∞ . Usingour notation, this means that the function g lies in R R . Geometrically, thiscondition also means that the chord to the graph of g on any fixed lengthinterval has an asymptotic zero slope. Only fixed length intervals whose leftendpoints are integers are to be considered if the condition reduces to requiringthat g ∈ R N . Our uniqueness and existence results show that this conditioncan actually be relaxed into g ∈ D N , which means that the chord to the graphof g on any interval of the form [ n , n + ] , n ∈ N ∗ , has an asymptotic zeroslope. The function g ( x ) = ln x is a typical example that shows, just as doesevery function whose derivative vanishes at infinity, that those functions neednot behave asymptotically like constant functions.It remains, however, that Krull-Webster’s asymptotic condition is ratherrestrictive. For instance, it is not satisfied for the functions g ( x ) = ln Γ ( x ) and g ( x ) = x ln x . To overcome this restriction, Rassias and Trif [72] proposeda modification of Webster’s results by considering solutions lying in K andreplacing the asymptotic condition with a more appropriate one. Specifically,they considered any function g : R + → R for which there exists a number a > such that lim t → ∞ g ( x + t ) − g ( t ) − x ln t = x ln a , for all x > . (84)It turns out that both functions g ( x ) = ln Γ ( x ) and g ( x ) = x ln x satisfy thisalternative condition.Let us now show that our asymptotic condition that g ∈ D R generalizes notonly Rassias and Trif’s (84) but also many other similar conditions. Proposition A.1.
Let ϕ : R + → R and suppose that g : R + → R has theproperty that, for each x > , g ( x + t ) − g ( t ) − x ϕ ( t ) → as t → ∞ . Then g lies in R R ⊂ D R . In particular, R R contains all the functions that satisfyRassias and Trif’s condition.Proof. For any t > and any g : R + → R , define the function ρ ϕt [ g ] : [ ∞ ) → R by ρ ϕt [ g ]( x ) = g ( x + t ) − g ( t ) − x ϕ ( t ) . Let also R ϕ R be the set of functions g : R + → R with the property that, for each x > , ρ ϕt [ g ]( x ) → as t → ∞ . Then we immediately see that ρ t [ g ]( x ) = ρ ϕt [ g ]( x ) − xρ ϕt [ g ]( ) , R ϕ R ⊆ R R . Now, if g satisfies Rassias and Trif’s condition,then it lies in the set ∪ a> R ϕ a R , where ϕ a ( x ) = ln ( ax ) , and hence it also lies in R R .Proposition A.1 can be generalized to R p R for any value of p > as follows. Proposition A.2.
Let p > be an integer, let ϕ
1, . . . , ϕ p − : R + → R , andsuppose that g : R + → R has the property that, for each x > , g ( x + t ) − g ( t ) − p − X j = (cid:0) xj (cid:1) ϕ j ( t ) → as t → ∞ . Then g lies in R p R ⊂ D p R .Proof. For any t > and any g : R + → R , define the function ρ ϕ t [ g ] : [ ∞ ) → R by ρ ϕ t [ g ]( x ) = g ( x + t ) − g ( t ) − p − X j = (cid:0) xj (cid:1) ϕ j ( t ) . Define also the functions ψ ϕ ,1 t [ g ] , . . . , ψ ϕ , pt [ g ] : [ ∞ ) → R recursively by ψ ϕ ,1 t [ g ] = ρ ϕ t [ g ] and ψ ϕ , j + t [ g ] = ψ ϕ , jt [ g ] − (cid:0) xj (cid:1) ψ ϕ , jt [ g ]( j ) , j =
1, . . . , p − Then, it is not difficult to see that ψ ϕ , jt [ g ]( x ) = ρ ϕ t [ g ]( x ) − j − X i = (cid:0) xi (cid:1) ( ∆ i g ( t ) − ϕ i ( t )) and hence ψ ϕ , pt [ g ] = ρ pt [ g ] . Thus, if the function g : R + → R has the propertythat, for each x > , ρ ϕ t [ g ]( x ) → as t → ∞ , then it lies in R p R . B On a question raised by Webster
Summary: We discuss conditions on function g to ensure both the uniqueness(up to an additive constant) and existence of solutions to the equation ∆f = g that lie in K p . A natural question raised by Webster [80, p. 606], and that we now extend toany value of p ∈ N , is the following: Find necessary and sufficient conditionson function g to ensure both the uniqueness (up to an additive constant)and existence of solutions lying in K p + (resp. K p − ) to the equation ∆f = g . Lemma 2.2(b) shows that a necessary condition for this to occur is that g ∈ K p − + (resp. g ∈ K p − − ). Also, our uniqueness and existence results showthat a sufficient condition is that g ∈ D p ∩ K p − (resp. g ∈ D p ∩ K p + ). It istempting to believe that this latter condition is also necessary. The followingtwo examples support this idea. 128a) Both functions ln Γ ( x ) and ln ( +
12 sin ( πx )) + ln Γ ( x ) are solutions to theequation ∆f = g that lie in K + , where g ( x ) = ln x does not lie in D ∪ K − .(b) Both functions x and x + sin ( πx ) are solutions to the equation ∆f = g that lie in K p + for any p ∈ N , where g ( x ) = x does not lie in D p ∪ K p − .Nevertheless, the following proposition shows that in general the conditionabove is not necessary. Proposition B.1.
There exists a function f ∈ C ∩ K such that(a) ∆f does not lie in D ∪ K , and(b) for any function ϕ ∈ K satisfying ∆ϕ = ∆f we have that f − ϕ isconstant.Proof. Let f ∈ K + be the function whose graph is the polygonal line throughthe points ( n , 4 n ) and ( n +
2, 4 n + ) for all n ∈ N . Thus the sequence n ∆f ( n ) is the -periodic sequence
2, 0, 0, 2, 2, 0, 0, 2, . . . and hence condition(a) holds. Now, let ϕ ∈ K be such that ∆ϕ = ∆f . Clearly, we must have ϕ ∈ K + . For the sake of a contradiction, suppose that the -periodic function ω = f − ϕ is not constant. That is, there exist < x < y such that ω ( x ) = ω ( y ) . There are two exclusive cases to consider.(a) Suppose that ω ( x ) < ω ( y ) . For large integer n , we then have ϕ ( y + n + ) − ϕ ( x + n + ) = ω ( x ) − ω ( y ) < (a) Suppose that ω ( x ) > ω ( y ) . For large integer n , we then have ϕ ( x + n + ) − ϕ ( y + n + ) = ω ( y ) − ω ( x ) < In both cases we reach a contradiction, and hence condition (b) holds.We note that the function f arising from Propositon B.1 is such that g = ∆f does not lie in D ∪ K . The following proposition shows that if the equation ∆f = g has a unique solution (up to an additive constant) and if g ∈ K p forsome p ∈ N , then necessarily g ∈ D p ∩ K p (see also Corollary 4.13). Proposition B.2.
Let g : R + → R and p ∈ N , and suppose that the sequence n ∆ p g ( n ) is eventually decreasing. Suppose also that there exists aunique (up to an additive constant) function f ∈ K p + satisfying the equation ∆f = g . Then g lies in D p N .Proof. For the sake of a contradiction, suppose that the assumptions are sat-isfied and that the sequence n ∆ p g ( n ) does not approach zero. Sincethis sequence is eventually nonnegative (because we eventually have ∆ p g = p + f > ), it must converge to a value C > . It follows that the function ~ g ( x ) = g ( x ) − C (cid:0) xp (cid:1) lies in D p ∩ K p − and hence Σ ~ g lies in K p + . Now, for any < τ < C/ ( π ) p , the functions f ( x ) = Σ ~ g ( x ) + C (cid:0) xp + (cid:1) , ϕ ( x ) = Σ ~ g ( x ) + C (cid:0) xp + (cid:1) + τ sin ( πx ) , lie in K p + ; indeed, we have D p + ( C (cid:0) xp + (cid:1) + τ sin ( πx )) > C + τ ( π ) p > Moreover, these functions are solutions to the equation ∆f = g and satisfy ϕ ( ) = f ( ) . This contradicts the uniqueness assumption. Remark
B.3 . We observe that if f and ϕ are solutions to ∆f = g , then for any x > and any p ∈ N ∗ , we have ∆ p f ( x ) > if and only if ∆ p ϕ ( x ) > . Indeed,suppose on the contrary that ∆ p f ( x ) > and ∆ p ϕ ( x ) < for some x > . Then ∆ p f ( x ) = ∆ p − g ( x ) = ∆ p ϕ ( x ) < a contradiction.Thus, Webster’s question still remains a very interesting open problem whosesolution would certainly shed light on the theory developed here.Regarding uniqueness issues only, the following two results due to John [42]are also worth mentioning. Generalizations of these results to higher convexityproperties would be welcome. Proposition B.4 (see [42]) . Let g : R + → R have the property that inf x ∈ R + g ( x ) = Then there is at most one (up to an additive constant) solution f to theequation ∆f = g that is increasing. Proposition B.5 (see [42]) . Let g : R + → R have the property that lim inf x → ∞ g ( x ) x = Then there is at most one (up to an additive constant) solution f to theequation ∆f = g that is convex. C Asymptotic behaviors and bracketing
Summary: We show that by considering higher and higher values of p in Theo-rem 6.5 we obtain closer and closer bounds for the Binet-like function J p + [ Σg ] . (cid:18) + x (cid:19) − Γ ( x ) √ π e − x x x − (cid:18) + x (cid:19) hold for any x > and that tighter inequalities can also be obtained by usingdifferent values of the integer p > in Theorem 6.5. In this appendix we showthat and how this feature applies in general to multiple log Γ -type functions.Let g ∈ C ∩ D p ∩ K p , where p = + deg p . By Theorem 6.5, for any x > such that g is p -convex or p -concave on [ x , ∞ ) we have the inequalities − G p | ∆ p g ( x ) | J p + [ Σg ]( x ) G p | ∆ p g ( x ) | . Let us now show how tighter inequalities can be obtained. For any r ∈ N ,define the functions α r [ Σg ] : R + → R and β r [ Σg ] : R + → R by the equations α r [ Σg ]( x ) = − G p + r (cid:12)(cid:12) ∆ p + r g ( x ) (cid:12)(cid:12) − p + r X j = p + G j ∆ j − g ( x ) ; β r [ Σg ]( x ) = G p + r (cid:12)(cid:12) ∆ p + r g ( x ) (cid:12)(cid:12) − p + r X j = p + G j ∆ j − g ( x ) . We immediately see that the equality α r [ Σg ]( x ) = β r [ Σg ]( x ) holds if andonly if ∆ p + r g ( x ) = . Also, by Theorem 6.5, if g ∈ K p + r and if x > is so that g is ( p + r ) -convex or ( p + r ) -concave on [ x , ∞ ) , then the following inequalitieshold: α r [ Σg ]( x ) J p + [ Σg ]( x ) β r [ Σg ]( x ) . The following proposition shows that these inequalities get tighter and tighteras the value of r increases. Proposition C.1.
Let g ∈ C ∩ D p ∩ K p + r + for some r ∈ N , where p = + deg g . Let x > be so that g | [ x , ∞ ) lies in K p + r ([ x , ∞ )) ∩ K p + r + ([ x , ∞ )) .Then, we have α r [ Σg ]( x ) α r + [ Σg ]( x ) β r + [ Σg ]( x ) β r [ Σg ]( x ) . These inequalities are strict if ∆ p + r g ( x + ) = .Proof. We already know that the central inequality holds. Now, using Corol-lary 4.14, we can assume that g is ( p + r ) -convex and ( p + r + ) -concave on [ x , ∞ ) ; the other case can be dealt with similarly. By Lemma 2.3, it follows that ∆ p + r g and ∆ p + r + g > on [ x , ∞ ) . Let us show that the first inequalityholds; the third one can be established similarly.We have two exclusive cases to consider.131 If G p + r + < , then ∆ r α r [ Σg ]( x ) = − G p + r + (cid:0) ∆ p + r + g ( x ) + ∆ p + r g ( x ) (cid:1) = − G p + r + ∆ p + r g ( x + ) .(cid:15) If G p + r + > , then ∆ r α r [ Σg ]( x ) = − G p + r ∆ p + r g ( x + )+ G p + r + (cid:0) ∆ p + r + g ( x ) − ∆ p + r g ( x ) (cid:1) . In both cases, we can see that ∆ r α r [ Σg ]( x ) > . Moreover, we have ∆ r α r [ Σg ]( x ) > if ∆ p + r g ( x + ) = .It is natural to wonder how the inequalities in Proposition C.1 behave as r → N ∞ . The following proposition, which is a reformulation of Proposition 6.8,answers this question and provides a series representation for J p + [ Σg ] . Proposition C.2.
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