aa r X i v : . [ m a t h . C V ] J a n ON THE DEFINITION OF HIGHER GAMMA FUNCTIONS
RICARDO P´EREZ-MARCO
Abstract.
We generalize our previous new definition of Euler Gamma functionto higher Gamma functions. With this unified approach, we characterize Barneshigher Gamma functions, Mellin Gamma functions, Barnes multiple Gamma func-tions, Jackson q -Gamma function, and Nishizawa higher q -Gamma functions. Thisapproach extends to more general functional equations. This generalization revealsthe multiplicative group structure of solutions of the functional equation that ap-pears as a cocycle equation. We also generalize Barnes hierarchy of higher Gammafunction and multiple Gamma functions. In this new approach, Barnes-Hurwitzzeta functions are no longer required for the definition of Barnes multiple Gammafunctions. This simplifies the classical definition, without the necessary analyticpreliminaries about the meromorphic extension of Barnes-Hurwitz zeta functions,and defines a larger class of Gamma functions. For some algebraic independenceconditions on the parameters, we have uniqueness of the solutions, which impliesthe coincidence of our multiple Gamma functions with Barnes multiple Gammafunctions. Introduction
The first result is a new characterization and definition of Euler Gamma functionthat was already presented in the article [19] dedicated to Euler Gamma function.We can develop in a natural way the classical formulas in the theory from this newdefinition. We denote the right half complex plane by C + = { s ∈ C ; Re s > } . Theorem 1.1.
There is one and only one finite order meromorphic function Γ( s ) , s ∈ C , without zeros nor poles in C + , with Γ(1) = 1 , Γ ′ (1) ∈ R , that satisfies thefunctional equation Γ( s + 1) = s Γ( s ) Definition 1.2 (Euler Gamma function) . The only solution to the above conditionsis the Euler Gamma function.
Mathematics Subject Classification.
Primary: 33B15. Secondary: 30D10, 30D15.
Key words and phrases.
Gamma function, Barnes Gamma function, Mellin Gamma functions,Jackson q -Gamma function, multiple gamma functions. Without the condition Γ ′ (1) ∈ R we don’t have uniqueness, but we have the fol-lowing result: Theorem 1.3.
Let f be a finite order meromorphic function in C , without zeros norpoles in C + , and satisfies the functional equation f ( s + 1) = s f ( s ) , then there exists a ∈ Z and b ∈ C such that f ( s ) = e πias + b Γ( s ) . Moreover, if f (1) = 1 then we have f ( s ) = e πias Γ( s ) . The proof can be found in [19] but we reproduce it here and as a preliminary resultfor the generalizations that are the core of this article. We refer to the companionarticle [19] for the various definitions of Euler Gamma function and the historicaldevelopment of the subject of Eulerian integrals. We strongly encourage the readerto study first [19], and also the bibliograhic notes in [20] before going into the gener-alizations that we develop in this article.In the proof we use the elementary theory of entire function and Weierstrass fac-torization that can be found in classical books as [5] (or in the Appendix of [19]).
Proof.
We prove existence and then uniqueness.
Existence:
If we have a function satisfying the previous conditions then its divisormust be contained in C − C + , and the functional equation implies that it has nozeros and only simple poles at the non-positive integers. We can construct such ameromorphic function g with such divisor, for example,(1) g ( s ) = s − ∞ Y n =1 (cid:16) sn (cid:17) − e s/n which converges since P n ≥ n − < + ∞ , and is of finite order. Now, we have that themeromorphic function g ( s +1) sg ( s ) has no zeros nor poles and it is of finite order (as ratioof finite order meromorphic functions), hence there exists a polynomial P such that g ( s + 1) sg ( s ) = e P ( s ) . Consider a polynomial Q such that(2) ∆ Q ( s ) = Q ( s + 1) − Q ( s ) = P ( s ) N THE DEFINITION OF HIGHER GAMMA FUNCTIONS 3
The polynomial Q is uniquely determined from P up to a constant, hence we canchoose Q such that e Q (0) = g (1) − . Now we have that Γ( s ) = e − Q ( s ) g ( s ) satisfies thefunctional equation and all the conditions. Uniqueness:
Consider a second solution f . Let F ( s ) = Γ( s ) /f ( s ). Then F is anentire function of finite order without zeros, hence we can write F ( s ) = exp A ( s ) forsome polynomial A . Moreover, the functional equation shows that F is Z -periodic.Hence, there exists an integer a ∈ Z , such that for any s ∈ C , A ( s + 1) = A ( s ) + 2 πia . It follows that A ( s ) = 2 πias + b for some b ∈ C . Since F (1) = 1, we have e b = 1.Since F ′ (1) ∈ R , and F ′ (1) = F ′ (1) /F (1) = 2 πia ∈ R we have a = 0, thus F isconstant, F ≡ f = Γ. (cid:3) Remarks. • Using the functional equation we can weaken the conditions and request onlythat the function is meromorphic only on C + with the corresponding finiteorder growth. We can also assume that it is only defined on a cone containingthe positive real axes, a vertical strip of width larger than 1, or in generalwith any region Ω which is a transitive region for the integer translations and f satisfies the finite order growth condition in Ω when s → + ∞ . Proposition 1.4.
Let Ω ⊂ C a domain such that for any s ∈ C there existsan integer n ( s ) ∈ Z such that s + n ( s ) ∈ Ω , and | n ( s ) | ≤ C | s | d , for someconstants C, d > depending only on Ω . Then any function ˜Γ satisfying afinite order estimate in Ω and the functional equation ˜Γ( s + 1) = s ˜Γ( s ) when s, s + 1 ∈ Ω , extends to a finite order meromorphic function on C .Proof. Let ˜Γ be such a function. Let Ω be corresponding region. Iteratingthe functional equation we get that ˜Γ extends meromorphically to the wholecomplex plane. Then, if g is the Weierstrass product (1) and Q a polynomialgiven by (2), the function h ( s ) = ˜Γ( s ) / ( e − Q ( s ) g ( s )) is a Z -periodic entirefunction. Since 1 / ( e − Q g ) is an entire function of finite order, we have in Ωthe finite order estimate for h . Using that | n ( s ) | ≤ C | s | d , we get that h is offinite order, hence ˜Γ is meromorphic and of finite order in the plane. (cid:3) • Assuming Γ real-analytic we get Γ ′ (1) ∈ R , but this last condition is muchweaker. Also, as it follows from the proof, we can replace this condition byΓ( a ) ∈ R for some a ∈ R − Z , or only request that Γ is asymptotically real,lim x ∈ R ,x → + ∞ Im Γ( x ) = 0. Without the condition Γ ′ (1) ∈ R the proof showsthat Γ is uniquely determined up to a factor e πiks with k ∈ Z . R. P´EREZ-MARCO General definition.
General definition and characterization.
We first need to recall the notionof “Left Located Divisor” (LLD) function that is useful in the theory of Poisson-Newton formula for finite order meromorphic functions ([14], [15]).
Definition 2.1 (LLD function) . A meromorphic function f in C is in the class LLD(Left Located Divisor) if f has no zeros nor poles in C + , i.e. Div( f ) ⊂ C − C + .The function is in the class CLD (Cone Located Divisor) if its divisor is containedin a closed cone in C − C + . The following Theorem is a generalization of Theorem 1.1 which results for thesimple LLD function f ( s ) = s . Theorem 2.2.
Let f be a real analytic LLD meromorphic function in C of finiteorder. There exists a unique function Γ f , the Gamma function associated to f , sat-isfying the following properties: (1) Γ f (1) = 1 , (2) Γ f ( s + 1) = f ( s )Γ f ( s ) , (3) Γ f is a meromorphic function of finite order, (4) Γ f is LLD, (5) Γ f is real analytic.If f is CLD then Γ f is CLD.Proof. The proof follows the same lines as the proof of Theorem 1.1. First, we provethat the functional equation (2) determines the divisor of Γ f , then we construct asolution using a Weierstrass product, and finally we prove the uniqueness. • Determination of the divisor.
As usual, we denote the divisor of f asDiv( f ) = X ρ n ρ ( f ) . ( ρ )where the sum is extended over ρ ∈ C and n ρ ( f ) is the multiplicity of the zero if ρ isa zero, the negative multiplicity of the pole if ρ is a pole, or n ρ ( f ) = 0 if ρ is neithera zero or pole. A divisor is said to be LLD, resp. CLD, if it is the divisor of a LLD,resp. CLD, function. N THE DEFINITION OF HIGHER GAMMA FUNCTIONS 5
Lemma 2.3. If Γ f is LLD and satisfies the functional equation (2), then the divisorof Γ f is Div(Γ f ) = − X ρ,k ≥ n ρ ( f ) · ( ρ − k ) where Div( f ) = X ρ n ρ ( f ) · ( ρ ) and (3) n ρ (Γ f ) = − | ρ | X k =0 n ρ + k ( f ) If the divisor
Div( f ) is LLD, resp. CLD, then Div(Γ f ) is LLD, resp. CLD. We allow ourselves the slight abuse of notation Div(Γ f ) to denote the divisor of apotential solution Γ f when we have not yet proved the existence of Γ f . Proof.
For any ρ ∈ C , the functional equation gives n ρ +1 (Γ f ) = n ρ ( f ) + n ρ (Γ f ) , or equivalently n ρ (Γ f ) = − n ρ ( f ) + n ρ +1 (Γ f ) , Hence, by induction, we have n ρ (Γ f ) = − m X k =0 n ρ + k ( f ) + n ρ + m (Γ f )and since Γ f is LLD, for m ≥ − Re ρ ≥ | ρ | we have n ρ + m (Γ f ) = 0, so(4) n ρ (Γ f ) = − | ρ | X k =0 n ρ + k ( f ) = − + ∞ X k =0 n ρ + k ( f )and we get, with ρ ′ = ρ + k ,Div(Γ f ) = − X k ≥ X ρ n ρ + k · ( ρ ) = − X ρ ′ ,k ≥ n ρ ′ · ( ρ ′ − k )which gives the formula for Div(Γ f ). (cid:3) R. P´EREZ-MARCO • Convergence exponent of a divisor.Definition 2.4.
The divisor of f has exponent of convergence α > if || Div( f ) || α = X ρ =0 | n ρ ( f ) | . | ρ | − α < + ∞ . We recall that a meromorphic function of finite order has a divisor with some finiteexponent of convergence. More precisely, if o ( f ) < + ∞ is the order of f , then forany ǫ > α = o ( f ) + ǫ is an exponent of convergence of its divisor. Proposition 2.5. If Div( f ) is LLD of finite order, then Div(Γ f ) given by Lemma2.3 is LLD and of finite order.Proof. We already known from Lemma 2.3 that Div(Γ f ) is LLD. We prove that if α is an exponent of convergence of Div( f ), then 2 α + 1 is an exponent of convergenceof Div(Γ f ) (we don’t try to be sharp here).First, observe that since α is an exponent of convergence for f , then || Div( f ) || α = X ρ =0 | n ρ ( f ) | . | ρ | − α ≤ C ( α ) < + ∞ , so we get | n ρ ( f ) | ≤ C ( α ) | ρ | α . Using equation (3) we have, for | ρ | ≥
1, and C ( α ) = C ( α )(1 + 2 α +1 / ( α + 1)), | n ρ (Γ f ) | ≤ | ρ | X k =0 | n ρ + k ( f ) | ≤ C ( α ) | ρ | X k =0 | ρ + k | α ≤ C ( α ) | ρ | X k =0 ( | ρ | + k ) α ≤ C ( α ) | ρ | α + Z | ρ | ( | ρ | + x ) α dx ! ≤ C ( α ) (cid:18) | ρ | α + 2 α +1 − α + 1 | ρ | α +1 (cid:19) ≤ C ( α ) | ρ | α +1 Therefore, we have || Div(Γ f ) || α +1 = X ρ, | ρ |≥ | n ρ (Γ f ) | . | ρ | − (2 α +1) < + ∞ . (cid:3) N THE DEFINITION OF HIGHER GAMMA FUNCTIONS 7
We prove a more precise result when f is in the class CLD. Proposition 2.6. If α > is an exponent of convergence for f in the class CLD,then Γ f is CLD and α + 1 is an exponent of convergence for Γ f . More precisely, thereexists a constant C > such that || Div(Γ f ) || α +1 ≤ C || Div( f ) || α +1 + Cα || Div( f ) || α Proof.
Lemma 2.3 proves that Γ f is CLD if we start with f CLD. Now, if f is CLD,there is a constant C > k ≥ ρ in the left cone (the constant C depends on the cone) | ρ − k | − ≤ C ( | ρ | + k ) − . Then we have, with ρ ′ = ρ + k , || Div(Γ f ) || β = X ρ =0 | n ρ (Γ f ) | . | ρ | − β = X ρ =0 | ρ | X k =0 | n ρ + k ( f ) | . | ρ | − β = X ρ ′ / ∈ N | n ρ ′ ( f ) | + ∞ X k =0 | ρ ′ − k | − β ≤ C X ρ ′ =0 | n ρ ′ ( f ) | + ∞ X k =0 ( | ρ ′ | + k ) − β = C X ρ ′ =0 | n ρ ′ ( f ) | . | ρ ′ | − β + C X ρ ′ =0 | n ρ ′ ( f ) | Z + ∞ ( | ρ ′ | + x ) − β dx ≤ C X ρ ′ =0 | n ρ ′ ( f ) | . | ρ ′ | − β + Cβ − X ρ ′ =0 | n ρ ′ ( f ) | . | ρ ′ | − β +1 = C || Div( f ) || β + Cβ − || Div( f ) || β − hence, for β = α + 1 the sum is converging and we prove the Lemma. (cid:3) • Existence of Γ f . Since f has finite order, the divisor of f has a finite convergence exponent. Hence,Div(Γ f ) determined by Lemma 2.3 has a finite exponent of convergence. Let d ≥ R. P´EREZ-MARCO be an integer that is an exponent of convergence for this divisor (the case d = 0 onlyoccurs for a finite divisor). We consider the Weierstrass product, g ( s ) = s − n ( f ) Y ρ =0 E d ( s/ρ ) n ρ (Γ f ) where E d ( x ) = (1 − x ) exp (cid:18) x + x . . . + x d d (cid:19) . Then g has order d and Div( g ) = Div(Γ f ). Therefore the meromorphic function g ( s + 1) f ( s ) g ( s )is of finite order and has no zeros nor poles. So, it is an entire function of finite orderwithout zeros. Therefore, there exists a polynomial φ such that(5) g ( s + 1) f ( s ) g ( s ) = e φ ( s ) There is a unique polynomial ψ such that ψ (0) = 0 and(6) ψ ( s + 1) − ψ ( s ) = φ ( s ) . We can obtain ψ directly by developing φ on the bases of falling factorial polynomials, s k = s ( s − . . . ( s − k + 1), that diagonalize the difference operator, ∆ s k = k s k − , φ ( s ) = n X k =0 a k k ! s k then ψ ( s ) = + ∞ X k =0 a k ( k + 1)! s k +1 . Now, considering a constant c such that e c = g (0) − the meromorphic function(7) Γ f ( s ) = e ψ ( s )+ c g ( s ) , satisfies Γ f (1) = 1 (condition (1)), the functional equation (2) and all the otherconditions in Theorem 2.2, and we have proved the existence. • Uniqueness of Γ f . Consider a second solution G . Let F ( s ) = Γ f ( s ) /G ( s ). Then F is an entire functionof finite order without zeros, hence we can write F ( s ) = exp A ( s ) for some polynomial N THE DEFINITION OF HIGHER GAMMA FUNCTIONS 9 A . Moreover, the functional equation shows that F is Z -periodic. Therefore, thereexists an integer a ∈ Z , such that for any s ∈ C , A ( s + 1) = A ( s ) + 2 πia . It follows that A ( s ) = 2 πias + b for some b ∈ C . Since F (1) = 1, we have e b = 1.Since F ′ (1) ∈ R , and F ′ (1) = F ′ (1) /F (1) = 2 πia ∈ R we have a = 0, thus F isconstant, F ≡ G = Γ f . (cid:3) Uniqueness results.
It is interesting to note, following the argument for unique-ness, that we can drop the normalisation condition (1) and the real-analyticity con-dition (5) and we obtain the following Theorem (this is similar to Theorem 1.3),
Theorem 2.7.
Let f be a LLD meromorphic function in C of finite order. Weconsider a function g satisfying (1) g ( s + 1) = f ( s ) g ( s ) , (2) g is a meromorphic function of finite order, (3) g is LLD,Then there is always a solution Γ f ( s ) and any other solution g is of the form g ( s ) = e πias + b Γ f ( s ) for some a ∈ Z and b ∈ C . If f is CLD then the solutions are CLD.Moreover, we have possible further normalizations: • If we add the condition g (1) = 1 , or g ( k ) = 1 for some k ∈ N ∗ , then allsolutions are of the form g ( s ) = e πias Γ f ( s ) . • If f − has a pole at and we add the condition Res s =0 g = 1 then all solutionsare of the form g ( s ) = e πias Γ f ( s ) . • If f has no zero at then we can add the condition g (0) = 1 and all solutionsare of the form g ( s ) = e πias Γ f ( s ) . • If we add the conditions g (1) = 1 and g ( ω ) ∈ R where ω ∈ R + − Q then g = Γ f is unique. • If we add the conditions g (1) = 1 and g ′ (1) ∈ R then the solution g = Γ f isunique. • If we add the hypothesis that f is real analytic and the condition that g is realanalytic then all solutions are of the form g ( s ) = c. Γ f ( s ) with c ∈ R ∗ .Proof. With the same proof as before we get the existence of a solution Γ f ( s ) andthat any other solution is of the form g ( s ) = e πias + b Γ f ( s ) (note that the constant 0 function is not LLD). For another solution g , the condition g ( k ) = 1 for k ∈ Z implies e b = 1, hence the first normalization result. For the second statement we observe thatRes s =0 g = e b Res s =0 Γ f hence e b = 1. The third statement is similar to the first one observing that g has nopole at s = 0. The fourth normalization condition forces b = 0 (first statement) and e πiaω = 1which implies a = 0 because ω is irrational. For the fifth statement, for a secondsolution we have, from g (1) = 1, g ( s ) = e πias Γ f ( s ). Differentiate and set s = 1, thenwe get g ′ (1) = 2 πiag (1) + (cid:0) Γ f (cid:1) ′ (1) = 2 πia + 1 ∈ R hence a = 0 and the solution is unique. For the last statement, g ( s ) = e πias + b Γ f ( s )and g and Γ f real analytic forces a = 0, and e b ∈ R ∗ . (cid:3) Example 2.8.
For f ( s ) = s and the conditions g real analytic and g (1) = 1, thisTheorem is just Theorem 1.1 and the only solution g ( s ) = Γ( s ) is Euler Gammafunction.Let ω ∈ C + and consider f ( s ) = ωs . Then g ( s ) = ω s Γ( s ) is a solution and all thesolutions are of the form g ( s ) = e πias + s log ω + b Γ( s ) =for a ∈ Z and b ∈ C (note that the choice of the branch of log ω is irrelevant).If ω ∈ C ∗ and we request g (1) = 1, then all solutions are of the form, with a ∈ Z ,(8) g ( s ) = e ( s − πia +log ω ) Γ( s )If ω ∈ R + , then f ( s ) = ωs is real analytic, and if we request g to be real analytic and g (1) = 1, then, taking the real branch of log ω , we must have a = 0 and(9) g ( s ) = e ( s −
1) log ω Γ( s ) Example 2.9.
Another particular example that is worth noting in this Theorem iswhen f ( s ) = e P ( s ) . Then the solutions are of the form g ( s ) = e Q k ( s ) where∆ Q k = P + 2 πik for k ∈ Z , where ∆ is the difference operator. This means that Q k ( s ) = Q ( s ) +2 πiks + b , where b ∈ C . If we want solutions normalized such that g (1) = 1 then e b = 1 and b ∈ πi Z . N THE DEFINITION OF HIGHER GAMMA FUNCTIONS 11
A continuity result.
We prove the continuity of the operator Γ : f Γ f forthe appropriate natural topology. Theorem 2.10.
Let ( f n ) n ≥ be a sequence of meromorphic functions with uniformlybounded convergence exponent α > and such that || Div( f n ) || α = X ρ ∈ Div( f n ) ,ρ =0 | n ρ || ρ | − α ≤ M < + ∞ for a uniform bound M > . We assume that the functions ( f n ) satisfy the hypothesisof Theorem 2.2 and that f n → f when n → + ∞ , where f is a meromorphic functionand the convergence is uniform on compact sets outside the poles of f . Then f hasconvergence exponent bounded by α > , || Div( f ) || α ≤ M < + ∞ and satisfies the hypothesis of Theorem 2.2, and also we have, uniformly outside thepoles, lim n → + ∞ Γ f n = Γ f Proof.
We can read the divisors Div( f n ) as an integer valued functions with discretesupport which are converging to Div( f ) uniformly on compact sets. By uniformboundedness of the sums || Div( f n ) || α = X ρ ∈ Div( f n ) ,ρ =0 | n ρ || ρ | − α we can pass to the limit and || Div( f ) || α = lim n → + ∞ || Div( f n ) || α ≤ M .
Therefore f has finite order. The class of LLD real analytic functions is closed. Theclass of functions satisfying the functional equation is also closed, hence f satisfiesthe hypothesis of Theorem 2.2, so Γ f is well defined.Now, since Div( f n ) → Div( f ), we have using Lemma 2.3 that Div(Γ f n ) → Div(Γ f ).On compact sets outside of the support of Div(Γ f ), the sequence of meromorphicfunctions (Γ f n ) n ≥ is uniformly bounded (otherwise we would have a subsequencewith poles out of the limit that would contradict the convergence of the divisor).Hence, we can extract converging subsequences. But any limit is identified by theuniqueness of Theorem 2.2, and we have convergence. (cid:3) Multiplicative group property.
Consider the space E of LLD finite ordermeromorphic functions in the plane. We have that E = [ n> E n where E n is the subgroup of meromorphic functions of order ≤ n . On E n we considerthe topology given by convergence of the divisor on compact sets and the convergenceof functions on compact sets outside the limit divisor. On E we consider the inductivetopology from the exhaustion by the E n spaces. Also E and E n are stable undermultiplication, and ( E , . ) and ( E n , . ) are multiplicative topological group. Considerthe closed subgroup E ⊂ E of real-analytic functions f normalized such that f (1) = 1. Theorem 2.11.
The map
Γ : E → E such that Γ( f ) = Γ f is an continuous injective group morphism.Proof. Continuity results from Theorem 2.10. We observe that fromΓ f ( s + 1) = f ( s )Γ f ( s )Γ g ( s + 1) = g ( s )Γ g ( s )we get Γ f ( s + 1)Γ g ( s + 1) = f ( s ) g ( s )Γ f ( s )Γ g ( s )and by uniqueness of Theorem 2.2 we getΓ f . Γ g = Γ fg . Also, if Γ f = 1, then directly from the functional equation we get that f = 1, andKer(Γ) = { } . (cid:3) This Theorem justifies using Euler Gamma function as building block of the generalsolution by decomposing along the divisor.
Remark.
Consider the shift operator T : E → E , f ( s ) T ( f ) = f ( s + 1) and the associatedmultiplicative cohomological equation in g with f given, T ( g ) .g − = f . We have proved that the cohomological equation can be solved in E by the groupmorphism Γ, g = Γ f . For f ∈ E α it can be solved in E α +1 . We observe a similarphenomenon of “loss of regularity” as in “Small Divisors” problems than in our settingcan be interpreted as “loss of transalgelbraicity”. N THE DEFINITION OF HIGHER GAMMA FUNCTIONS 13 Application: Barnes higher Gamma functions.
We generalize the classical hierarchy of Barnes Gamma functions.
Definition 3.1.
Let f be a real analytic LLD meromorphic function of finite ordersuch that f (1) = 1 . The higher Gamma functions associated to f is a family (Γ fN ) N ≥ satisfying the following properties: (1) Γ f ( s ) = f ( s ) , (2) Γ fN (1) = 1 , (3) Γ fN +1 ( s + 1) = Γ fN ( s ) − Γ fN +1 ( s ) , for N ≥ , (4) Γ fN is a meromorphic function of finite order, (5) Γ fN is LLD, (6) Γ fN is real analytic. Theorem 3.2.
Let f be a real analytic LLD meromorphic function of finite ordersuch that f (1) = 1 . There exists a unique family of higher Gamma functions (Γ fN ) N associated to f . If f is CLD then the Γ fN are CLD.Proof. We set Γ f ( s ) = f ( s ), and for N ≥
0, the function Γ fN +1 is constructed from1 / Γ fN using Theorem 2.2, and is unique. (cid:3) The uniqueness property implies the following multiplicative group morphism prop-erty:
Corollary 3.3.
For N ≥ , we consider the map Γ N : E → E defined by Γ N ( f ) =Γ fN . Then Γ N is a continuous injective group morphism.Proof. Given f, g ∈ E , it is clear that the sequence of functions Γ fN . Γ gN satisfy allthe properties of higher Gamma functions associated to f g , hence, by uniqueness, wehave Γ fgN = Γ fN Γ gN , hence the group morphism property. The kernel is reduced to theconstant function 1 by uniqueness, hence the injectivity. The continuity follows asbefore from Theorem 2.10. (cid:3) Definition 3.4 (Barnes higher Gamma functons Γ N ) . The higher Gamma functionsassociated to f ( s ) = s is the family of higher Barnes Gamma functions (Γ N ) N ≥ , and Γ is Euler Gamma function. Note that Vign´eras’ normalization (1979, [28]) is slightly different and defines (for f ( s ) = s ) a hierarchy of functions ( G fN ) N ≥ as in Definition 3.1 but with the functionalequation replaced by G fN +1 ( s + 1) = G fN ( s ) G fN +1 ( s ) We have a simple direct relation between the two hierarchies G fN = (Γ fN ) ( − N +1 . For f ( s ) = s we obtain G f = G which is Barnes G -function (Barnes, 1900, [3]). Theconvention in Definition 3.1 is compatible with Barnes multiple Gamma functionsthat generalize the (Γ N ) (Barnes, 1904, [4], and Section 6). Proposition 3.5.
The higher Barnes Gamma function Γ N is CLD of order N , and Div(Γ N ) = − + ∞ X n =0 (cid:18) n + N − N − (cid:19) . ( − n ) Proof.
The function Γ N is in the class CLD by induction since f is in this class. Any α > f ( s ) = s , so by Proposition 2.6 we have byinduction that any α > N is exponent of convergence for Γ N . We can check thisdirectly using the formula for the divisor that follows by induction from Lemma 2.3and the combinatorial identity (cid:18) n + NN (cid:19) = n X k =0 (cid:18) k + N − N − (cid:19) If we write the Weierstrass factorization and Q N denotes the Weierstrass polynomial,we have that deg Q = 1, and by induction the same proof gives that deg Q N = N . (cid:3) When we drop the real analyticity condition, there is no longer uniqueness, but wecan prove the following Theorem,
Theorem 3.6.
Let f be a LLD meromorphic function of finite order such that f (1) =1 . Consider a family ( g fN ) N ≥ satisfying the following properties: (1) g f ( s ) = f ( s ) , (2) g fN (1) = 1 , (3) g fN +1 ( s + 1) = g fN ( s ) − g fN +1 ( s ) , for N ≥ , (4) g fN is a meromorphic function of finite order, (5) g fN is LLD,Then there exists an integer sequence ( a k ) k ≥ , such that g fN ( s ) = exp πi N X k =0 a N − k (cid:18) sk (cid:19)! Γ fN ( s ) Proof.
This follows by induction from Theorem 2.7. We can also give a direct argu-ment using the group structure. For any solution ( g fN ) N ≥ , the functions h fN = Γ fN /g fN N THE DEFINITION OF HIGHER GAMMA FUNCTIONS 15 are solution for f = 1. The case f = 1 is easily resolved. By induction, the solutionshave no zeros nor poles, and finite order, so we have h fN ( s ) = e πiA N ( s ) where the ( A N ) N ≥ is a sequence of polynomials satisfying∆ A N +1 = − A N and A ( s ) = a ∈ Z . The difference equation and the sequence a N = ( − N A N (0)determines the sequence of polynomials ( A N ) N ≥ that are given by the explicit formula A N ( s ) = N X k =0 a N − k (cid:18) sk (cid:19) (cid:3) Application: Jackson q -Gamma function. For 0 < q <
1, Jackson (1905, [10], [11]) (see also the precursor work by Halphen[7], vol. 1, p. 240; and H¨older [8]) defined the q -Gamma function Γ q by the productformula Γ q ( s ) = ( q ; q ) ∞ ( q s ; q ) ∞ (1 − q ) − s where the ∞ -Pochhammer symbol is( z ; q ) ∞ = + ∞ Y k =0 (1 − zq k ) . The q -Gamma function satisfies the functional equationΓ q ( s + 1) = 1 − q s − q Γ q ( s )and Euler Gamma function appears as the limit when q → s ) = lim q → − Γ q ( s )Askey ([1], 1980) proved a q -analog of the Bohr-Mollerup theorem characterizing Γ q by its functional equation, the normalization Γ q (1) = 1, and the real log-convexity ofΓ q . It is natural to investigate if we can use our approach. The answer is affirmativeas shows the next Theorem. Theorem 4.1.
The q -Gamma function is the only real analytic, finite order mero-morphic function such that Γ q (1) = 1 and satisfying the functional equation, Γ q ( s + 1) = 1 − q s − q Γ q ( s ) Proof.
This is an application of our general Theorem 2.2 with f ( s ) = 1 − q s − q which is an order 1 real analytic function in the class LLD (but not CLD), f (1) = 1,and Div( f ) = X k ∈ Z . (cid:18) πik log q (cid:19) . (cid:3) An application of the continuity Theorem 2.10 shows:
Proposition 4.2.
We have lim q → − Γ q = Γ uniformly on compact sets of C .Proof. Uniformly on compact sets of C we havelim q → − − q s − q = s and we use Theorem 2.10. (cid:3) Nishiwaza (1996, [16]) has defined the q -analog Γ N,q of Barnes higher Gammafunctions Γ N following the Bohr-Mollerup approach. With our methods we can obtainNishiwaza’s Γ N,q functions directly from the higher hierarchy generated by f usingDefinition 3.1 and Theorem 3.2 using the uniqueness of the solution. Theorem 4.3.
Nishiwaza’s higher q -Gamma functions Γ N,q are obtained by the higherhierarchy from Theorem 3.2 Γ N,q = Γ fN associated to the real analytic function f ( s ) = 1 − q s − q . Application: Mellin Gamma functions.
Mellin (1897, [13]) considered general Gamma functions satisfying the functionalequation F ( s + 1) = R ( s ) F ( s )where R is a rational function. He constructs solutions by using Euler Gamma func-tion as building block along the divisor. An application of the extension of our general N THE DEFINITION OF HIGHER GAMMA FUNCTIONS 17
Theorem 2.7, and the group structure Theorem 2.11, gives the precise existence char-acterization of Mellin Gamma functions.
Definition 5.1.
A meromorphic function f is LLD at infinite if f ( s + a ) is LLD forsome a ∈ R . Since Div( f ( s + a )) = Div( f ) − a this means that the divisor of f is in some lefthalf plane (not necessarily C + ). Theorem 5.2.
Let R be a rational function, R ( s ) = a ( s − α ) . . . ( s − α n )( s − β ) . . . ( s − β m ) where a ∈ C ∗ , and ( α k ) and ( β k ) are the zeros, resp. the poles, of R counted withmultiplicity.Consider the finite order meromorphic functions, LLD at infinite, that are solutionsof the functional equation (10) F ( s + 1) = R ( s ) F ( s ) . They are of the form F ( s ) = a s Γ( s − α ) . . . Γ( s − α n )Γ( s − β ) . . . Γ( s − β m ) e πiks for some k ∈ Z .In particular, if R (1) = 1 and R is real analytic there is only one real analyticsolution such that F (1) = 1 .Proof. Let α be a zero or pole. We consider the linear function f α ( s ) = s − α and asolution Γ f α to F α ( s + 1) = f α ( s ) F α ( s + 1) . Also a s is a solution to F ( s + 1) = aF ( s ). Then, Theorem 2.7 and the group structureof the solutions, Theorem 2.11, shows that the general solutions of the functionalequation (10) are of the form F ( s ) = a s e πins Γ f α ( s ) e πik s . . . Γ f αn ( s ) e πik n s Γ f β ( s ) e πil s . . . Γ f βm ( s ) e πil m s = a s Γ f α ( s ) . . . Γ f αn ( s )Γ f β ( s ) . . . Γ f βm ( s ) e πiks where n, k , . . . , k n , l , . . . , l m ∈ Z , and k = n + k + . . . k n + l + . . . + l m . We finish the proof by observing that we can take Γ f α ( s ) = Γ( s − α ). When R isreal analytic, a ∈ R ∗ , the set of roots ( α j ) and poles ( β j ) are self-conjugated, and wemust have k = 0 to have F real analytic. (cid:3) Considering a LLD rational function R , real analytic and such that R (1) = 1, wecan define the unique associated higher Gamma functions (Γ RN ) N ≥ given by Theorem3.2. These higher Mellin Gamma functions do not seem to appear in the literature.6. Application: Barnes multiple Gamma functions.
For N ≥ ω = ( ω , . . . , ω n ) ∈ C n + , Barnes multiple Gammafunctions Γ( s | ω , . . . , ω N ) = Γ( s | ω ) are a generalization by Barnes (1904, [4]) ofBarnes higher Gamma functions Γ N studied in section 3. When ω = . . . = ω N = 1we recover Γ N as Γ N ( s ) = Γ( s | , . . . , ω , . . . , ω n all belong toa half plane limited by a line through the origin ([4] p.387). This situation that canbe reduced to our case by a rotation. Also, he assumes dim Q ( ω , . . . , ω N ) ≥ G that he studiedpreviously, although this condition is not the appropriate one. Barnes defines thesemultiple Gamma functions `a la Lerch . First, Barnes defines the Barnes-Hurwitz zetafunctions, a multiple version of Hurwitz zeta function, as ζ ( t, s | ω , . . . , ω N ) = X k ,...k N ≥ ( s + k ω + . . . + k N ω N ) − t , which is converging for Re s > N , and symmetric on ω , . . . , ω N . This multiple zetafunction reduces to Hurwitz zeta function for N = 1 (Hurwitz, 1882, [9]). Its analyticcontinuation and Lerch formula (Lerch, 1894, [12])(11) log Γ( s ) = (cid:20) ∂∂t ζ ( t, s ) (cid:21) t =0 − ζ ′ (0)allows to define Euler Gamma function. Barnes generalizes this approach and heshows, using a Hankel type integral, that ζ ( s, t | ω , . . . , ω N ) has a meromorphic exten-sion in ( s, t ). Then he definesΓ B ( s | ω ) = ρ N ( ω ) exp (cid:18)(cid:20) ∂∂t ζ ( t, s | ω ) (cid:21) t =0 (cid:19) where ρ N ( ω ) is Barnes modular function, and is defined to provide the normalizationsuch that Γ B ( s | ω ) has residue 1 at s = 0, N THE DEFINITION OF HIGHER GAMMA FUNCTIONS 19 (12) Res s =0 Γ B ( s | ω ) = lim s → s Γ B ( s | ω ) = 1From the definition we get that both ρ N ( ω ) and Γ B ( s | ω ) are necessarily symmetricon ω , . . . , ω N . Note that for Euler Gamma function, because of the form of thefunctional equation, the normalization Γ(1) = 1 is equivalent to Res s =0 Γ = 1. Ingeneral, for Γ f the normalization Γ f (1) = 1 is equivalent toRes s =0 Γ f = Res s =0 f − . For Barnes higher Gamma functions Γ N discussed in section 3, we see that the nor-malization Γ N (1) = 1 is equivalent to Res s =0 Γ N = 1 when we make s → N +1 ( s + 1) = ( s Γ N ( s )) − s Γ N +1 ( s )we get Γ N +1 (1) = Res s =0 Γ N +1 = 1 . and the result follows by induction.Barnes ([4], p.397) observes that log ρ ( ω ) plays the role of Stirling’s constant of theasymptotic expansion when k → + ∞ of the divergent sum X ω ∈ Ω ∗ , | ω |≤ k log | ω | where Ω ∗ = N .ω + N .ω + . . . + N .ω N − { } . In this way, log ρ ( ω ) can also be defined.Later applications to Number Theory by Shintani in the 70’s of Barnes multipleGamma functions (1976,[25], [26], [27]), and modern presentations (Ruijsenaars, [22]),drop Barnes normalization. They define multiple Gamma functions directly by theformula Γ( s | ω ) = exp (cid:18)(cid:20) ∂∂t ζ ( t, s | ω ) (cid:21) t =0 (cid:19) We keep Shintani’s normalization that has become the usual one in recent articles.This modern normalization has the advantage to yield a simpler functional equationnot involving Barnes modular function ρ ( ω ). For ω = ω k , we denote ˆ ω the N − ω removing the k -th coordinate. Then we have thefollowing ladder functional equation for the zeta function,(13) ζ ( t, s + ω | ω ) − ζ ( t, s | ω ) = − ζ ( t, s | ˆ ω )where we start with ζ ( t, s |∅ ) = s − t . From the zeta function functional equation we get the functional equation for themultiple Gamma functions,(14) Γ( s + ω | ω ) = Γ( s | ˆ ω ) − Γ( s | ω ) with the convention Γ( s |∅ ) = s . Note that the functional equation for Barnes nor-malized multiple Gamma functions is different:(15) Γ B ( s + ω | ω ) = ρ ( ˆ ω )Γ B ( s | ˆ ω ) − Γ B ( s | ω ) . Example 6.1.
For N = 1, Γ( s | ω ) can be computed explicitly from Euler Gammafunction (see [24], p.203). Lemma 6.2.
We have Γ( s | ω ) = (2 π ) − / e ( sω − ) log ω Γ (cid:16) sω (cid:17) ρ ( ω ) = r ω π and therefore Γ B ( s | ω ) = r πω Γ( s | ω ) = e ( sω − ) log ω Γ (cid:16) sω (cid:17) and Γ( ω | ω ) = r ω π Res s =0 Γ( s | ω ) = r ω π Γ B ( ω | ω ) = 1 Res s =0 Γ B ( s | ω ) = 1 In particular, Γ( s |
1) = Γ( s ) √ π Γ B ( s |
1) = Γ( s ) Proof.
For ω = 1, ζ ( t, s |
1) = ζ ( t, s ) is the original Hurwitz zeta function that gener-alizes Riemann zeta function ζ ( t ) = ζ ( t, ζ ( t, s ) = X k ≥ ( s + k ) − t . Making t = 0 in the first formula from Lemma 3.18 from [19] we have the classicalresult (see also [29] p.267)(16) ζ (0 , s ) = 12 − s . N THE DEFINITION OF HIGHER GAMMA FUNCTIONS 21
Observe now that we have ζ ( t, s | ω ) = ω − t ζ (cid:0) t, sω (cid:1) , hence ∂∂t ζ ( t, s | ω ) = − (log ω ) ω − t ζ (cid:16) t, sω (cid:17) + ω − t ∂∂t ζ ( t, s )and making t = 0, using formula (16) and Lerch formula (11), we getlog Γ( s | ω ) = (cid:18) sω − (cid:19) log ω + log Γ (cid:16) sω (cid:17) + ζ ′ (0) . Now, ζ ′ (0) = − log(2 π ) gives the first formula. Then using this formula we getRes s =0 Γ( s | ω ) = lim s → s Γ( s | ω ) = (2 π ) − / e − log ω ω = r ω π . (cid:3) For N ≥ s | ω ), which are not generated fromEuler Gamma function. For example for N = 2, if ω and ω are Q -independent weget new transcendentals. When the parameters are Q -dependent then Γ( s | ω ω ) canbe expressed from Barnes G -function, G = Γ − .From the functional equations, and from our point of view, it is natural to aim tocharacterize Γ( s | ω ) by solving a tower of difference equations corresponding to thesequence ( ω k ) ≤ k ≤ n . Our approach leads to a new definition, not needing Barnes-Hurwitz zeta functions. We start by considering real analytic multiple zeta functionsthat are those relevant in Shintani’s applications to real quadratic number fields (1978,[26]). The following result follows from Theorem 2.7. Theorem 6.3.
Let ω ∈ R + . Let f be a real analytic LLD meromorphic functionin C of finite order. There exists a unique function Γ f ( s | ω ) satisfying the followingproperties: (1) Γ f (1 | ω ) = 1 , (2) Γ f ( s + ω | ω ) = f ( s )Γ f ( s | ω ) , (3) Γ f ( s | ω ) is a meromorphic function of finite order, (4) Γ f ( s | ω ) is LLD, (5) Γ f is real analytic.If f is CLD then Γ f is CLD.If we drop condition (1) then Γ f ( s | ω ) is unique up to multiplication by a constant c ∈ R ∗ .If Res s =0 f − = 1 , we can replace condition (1) by the condition Res s =0 Γ f = 1 . Proof.
We make the change of variables t = ω − s . The application of Theorem 2.7to the real analytic function h ( t ) = f ( ωt ) gives a unique real analytic solution Γ h ( t )such that Γ h (1) = 1 and Γ h ( t + 1) = h ( t )Γ h ( t ) . If we set Γ f ( s | ω ) = Γ h ( ω − s ), this equation becomesΓ f ( s + ω | ω ) = Γ h ( ω − s + 1) = h ( ω − s )Γ h ( ω − s ) = f ( s )Γ f ( s | ω )and Γ f ( s | ω ) satisfies all conditions. Furthermore, Γ f ( s | ω ) is unique from the unique-ness of Γ h that follows from the last uniqueness condition in Theorem 2.7. In view ofthis uniqueness result, the two last statement are clear. Also if f is CDL then Γ f ( s | ω )is CDL. (cid:3) Example 6.4.
For f ( s ) = s the proof gives h ( t ) = ωt and a solution Γ f ( s | ω ) =Γ h (cid:0) tω (cid:1) . The condition Γ f (1 | ω ) = 1 is equivalent to Γ h (cid:0) ω (cid:1) = 1, then according toExample 2.8 there is a unique real analytic solutionΓ h ( t ) = e ( t −
1) log ω Γ( t )Γ( ω − )and it follows that Γ f ( s | ω ) = e ( sω − ) log ω Γ (cid:0) sω (cid:1) Γ( ω − )Therefore, by uniqueness of the normalization,Γ B ( s | ω ) = Γ( ω − )Γ f ( s | ω )and we recover the formula for Γ B ( s | ω ) from Lemma 6.2Γ B ( s | ω ) = e ( sω − ) log ω Γ (cid:16) sω (cid:17) Then the formula for Γ( s | ω ) follows fromΓ( s | ω ) = r ω π Γ B ( s | ω ) = (2 π ) − / e ( sω − ) log ω Γ (cid:16) sω (cid:17) . We have established,
Proposition 6.5.
For f ( s ) = s we have Γ( s | ω ) = r ω π Γ( ω − ) Γ f ( s | ω ) where Γ f ( s | ω ) is the unique solution in Theorem 6.3. Using similar ideas, the general version of Theorem 2.7 for ω ∈ C + and withoutthe hypothesis of f being real analytic is the following: N THE DEFINITION OF HIGHER GAMMA FUNCTIONS 23
Theorem 6.6.
Let ω ∈ C + . Let f be a LLD meromorphic function in C of finiteorder. We consider a function g satisfying (1) g (1) = 1 , (2) g ( s + ω ) = f ( s ) g ( s ) , (3) g is a meromorphic function of finite order, (4) g is LLD,Then there is a solution Γ f ( s | ω ) . Any other solution g is of the form g ( s ) = e πia s − ω Γ f ( s | ω ) for some a ∈ Z .If we remove condition (1) then all solutions are of the form the form g ( s ) = e b +2 πia sω Γ f ( s | ω ) for some a ∈ Z and b ∈ C .Proof. As before, we make the change of variables t = ω − s and apply Theorem 2.7 tothe function h ( t ) = f ( ωt ) gives an unconditional solution Γ f ( s | ω ) = Γ h ( t ) / Γ h ( ω − ).From the general uniqueness statement in 2.7 we know that all the other solutionsremoving condition (1) are of the form g ( s ) = e πia sω + b Γ f ( s | ω ) for some a ∈ Z and b ∈ C . Condition (1) is then equivalent to 2 πia/ω + b = 2 πik with k ∈ Z , hence thegeneral form. (cid:3) Therefore, in general for ω ∈ C ∗ , Γ f is not uniquely determined, but its values on1 + Z .ω are well determined. More precisely, we have Proposition 6.7.
The values taken by solutions at the points kω for k ∈ Z areuniquely determined and do not depend on the solution chosen.If ω ∈ C + , any solution g is uniquely determined by Im g ′ (1) , in particular, if f isreal analytic then there is a unique real analytic solution.If dim Q (1 , ω ) = 2 , any solution g is uniquely determined by its value g ( k ) for someinteger k ≥ .Proof. From the functional equation we have g (1 + kω ) = g (1) k − Y j =0 f (1 + jω ) = k − Y j =0 f (1 + jω )hence the first claim.Now, consider two solutions g and g such that Im g ′ (1) = Im g ′ (1). Since they areof the form g j ( s ) = e πia j s − ω Γ f ( s | ω ) for some a j ∈ Z , taking logarithmic derivativeswe have g ′ j (1) = g ′ j (1) g j (1) = 2 πi a j ω + (cid:0) Γ f (cid:1) ′ (1 | ω )Γ f (1 | ω ) = 2 πi a j ω + (cid:0) Γ f (cid:1) ′ (1 | ω ) hence g ′ (1) − g ′ (1) = 2 πi a − a ω ∈ R and the condition ω ∈ C + forces a = a .Now assume dim Q (1 , ω ) = 2 and consider two solutions g and g such that g ( k ) = g ( k ) for some integer k ≥
2. Then, since s = k is neither a zero nor a pole, we havefor some l ∈ Z g ( k ) g ( k ) = e π ( k − a − a ω = 1thus, for some integer l ∈ Z , we have( k − a − a ) − lω = 0and, by Q -independence, we must have l = 0 and ( k − a − a ) = 0, thus, since k ≥ a = a and g = g . (cid:3) General Multiple Gamma Hierarchies.
Now, we can iterate Theorem 6.3 to define new real-analytic multiple Gammafunction corresponding to f and positive real parameters ω = ( ω , . . . , ω N ) ∈ R N + For a sequence of parameters ω = ( ω , ω , . . . ) ∈ C ∞ + , we can now define a general-ization of Barnes multiple Gamma hierarchy. We denote ω N = ( ω , . . . , ω N ) ∈ C N + . Definition 6.8 (General Multiple Gamma Hierarchy) . Let ω = ( ω , ω , . . . ) ∈ C ∞ + and f be a LLD meromorphic function in C of finite order. A general multiple Gammahierarchy (Γ fN ( s | ω N )) N ≥ associated to f is a sequence of functions satisfying: (1) Γ f ( s ) = f ( s ) , (2) Γ fN +1 ( s + ω N +1 | ω N +1 ) = Γ fN ( s | ω N ) − Γ fN +1 ( s | ω N +1 ) , for N ≥ , (3) Γ fN ( s | ω N ) is a meromorphic function of finite order, (4) Γ fN ( s | ω N ) is LLD. Next we show that, with some simple normalization, General Multiple GammaHierarchies are unique for real parameters and f real analytic. Theorem 6.9.
Let ω = ( ω , ω , . . . ) ∈ R ∞ + and f a real analytic LLD meromorphicfunction of finite order, such that f (1) = 1 . There exists a unique General MultipleGamma Hierarchy (Γ fN ( s | ω N )) N ≥ associated to f , and normalized such that Γ fN (1 | ω N ) = 1 . If f is CLD then the Γ fN ( s | ω N ) are CLD. N THE DEFINITION OF HIGHER GAMMA FUNCTIONS 25
Proof.
The existence and uniqueness is proved by induction on N ≥
0. For N = 0,Γ f ( s ) = f ( s ). We assume that the result has been proved for N ≥
0. Then weconstruct Γ fN +1 ( s | ω N +1 ) by using Theorem 6.3 using the function f = Γ fN ( s | ω N ) − . (cid:3) The particular case f ( s ) = s , using uniqueness, yields Barnes multiple Gammafunctions for real parameters ω . Definition 6.10 (Barnes multiple Gamma functions) . For ω = ( ω , ω , . . . ) ∈ R ∞ + theGeneral Multiple Gamma Hierarchy associated to f ( s ) = s is Barnes Multiple GammaHierarchy (Γ fN ( s | ω N )) N ≥ with the normalization Γ fN (1 | ω N ) = 1 . We simplify thenotation and we denote Γ fN ( s | ω N ) = Γ( s | ω N ) . We observe that since the Barnes multiple Gamma functions Γ( s | ω N ) are symmet-ric on the real parameters ( ω , . . . , ω N ) then, by uniqueness, the solutions of Theorem6.9 for f ( s ) = s must also be symmetric on the parameters. This is general when wecan define the Gamma functions `a la Lerch, including the case of complex parameters ω = ( ω , ω , . . . ) ∈ C ∞ + . Consider f a real analytic LLD meromorphic function offinite order, such that, f (1) = 1and Re f ( s ) > s ∈ C + . These conditions are sufficient to define f ( s ) − t for s ∈ C + by taking the principal branch of log in C + , f ( s ) − t = exp( − t log f ( s )). Weassume that the multiple Barnes-Hurwitz multiple zeta function associated to f , ζ f ( t, s | ω , . . . , ω N ) = X k ,...k N ≥ f ( s + k ω + . . . + k N ω N ) − t , is well defined and holomorphic in a right half plane Re t > t for all s ∈ C + , and hasa meromorphic extension to t ∈ C . We define Γ f ( s |∅ ) = f ( s ) − t , and, `a la Lerch, for s ∈ C + , Γ fL ( s | ω N ) = exp (cid:20) ∂∂t ζ f ( t, s | ω N ) (cid:21) t =0 − (cid:20) ∂∂t ζ f ( t, s | ω N ) (cid:21) t =0 ,s =1 ! Note that we have normalized these functions such that Γ fL (1 | ω N ) = 1. By con-struction, these functions are obviously symmetric on the parameters ω , . . . , ω N . Asbefore, these functions satisfy the functional equations,(17) Γ fL ( s + ω N | ω N ) = Γ fL ( s | ω N − ) − Γ fL ( s | ω N )which show that they have a meromorphic extension to all s ∈ C . now, using theuniqueness from Theorem 6.9 we get for real parameters: Theorem 6.11.
Let ω = ( ω , ω , . . . ) ∈ R ∞ + . When Γ fL ( s | ω N ) is well defined, wehave Γ fL ( s | ω N ) = Γ fN ( s | ω N ) where the (Γ fN ( s | ω N )) N ≥ are the solutions of Theorem 6.9. Corollary 6.12.
Let ω = ( ω , ω , . . . ) ∈ R ∞ + . The Barnes multiple Gamma hier-archy defined by Theorem 6.9, Γ fN ( s | ω N ) are symmetric on the parameters ω N =( ω , ω , . . . , ω N ) . We should note that our definition of the hierarchies using the functional equationis more general than Barne’s definition `a la Lerch, since we need conditions on f sothat the multiple f -Barnes-Hurwitz zeta function is well defined and holomorphic ina half plane. If we don’t add the normalization conditionΓ fN (1 | ω N ) = 1then there are solutions that are non-symmetric on the parameters. As we see next,this is even more evident for complex parameters since in that case, without furtherhypothesis, there is no symmetry on the parameters ω . This shows that our functionalequation approach defines a larger class of functions.We observe also that the existence and uniqueness of Theorem 6.9 implies the mor-phism property. Let E R be the multiplicative group of real-analytic LLD meromorphicfunctions of finite order and E R = \ n ≥ E R n and E R the subgroup of functions f such that f (1) = 1. With the same arguments asbefore, we have Theorem 6.13.
For ω = ( ω , ω , . . . ) ∈ R ∞ + and N ≥ , we consider the map Γ N ( ω N ) : E R → E R defined by Γ N ( ω N )( f ) = Γ fN ( . | ω N ) . Then Γ N ( ω N ) is a continuous injective groupmorphism. Complex parameters.
We study now the non-real-analytic case for complex parameters ω , . . . , ω N ∈ C + . In general we don’t have uniqueness as in Theorem 6.9. We consider f a LLDmeromorphic function in C of finite order with f (1) = 1 and study the questionof existence and uniqueness of a general multiple Gamma functions hierarchy as inDefinition 6.8 with the normalizationΓ fN (1 | ω N ) = 1 . N THE DEFINITION OF HIGHER GAMMA FUNCTIONS 27
We have the following result without the real analyticity condition:
Theorem 6.14.
Let ω = ( ω , ω , . . . ) ∈ C ∞ + and f a LLD meromorphic functionof finite order such that f (1) = 1 . There exists General Multiple Gamma Hierarchy (Γ fN ( s | ω N )) N ≥ associated to f , and for any other hierarchy (˜Γ fN ( s | ω N )) N ≥ thereexists a sequence of polynomials ( P N ) N ≥ such that ˜Γ fN ( s | ω N ) = exp (2 πiP N ( s )) Γ f N ( s | ω N ) with P N (1) ∈ Z , P is a constant integer, and for N ≥ we have ∆ ω N +1 P N +1 = − P N where ∆ ω is the ω -difference operator ∆ ω P = P ( s + ω ) − P ( s ) . The space of polyno-mials P N is isomorphic to Z N +1 .If the functions f is CLD then Γ f N ( s | ω N ) and all the other solutions are CLD.Proof. For the existence result, it is the same proof by induction as for Theorem 6.9(without the normalization condition) and using Theorem 6.6. If a second solution(˜Γ fN ( s | ω N )) N ≥ exists, then (˜Γ fN ( s | ω N ) / Γ fN ( s | ω N )) N ≥ is a solution of the problemfor the constant function f ( s ) = 1. The solution for f ( s ) = 1 has no divisor and isof finite order, hence they are of the form exp( P N ) where P N are polynomial whichsatisfy the above difference equations. Next in what follows, we discuss uniquenessconditions and the structure of the general polynomials P N will become clear. (cid:3) We observe that the integer sequence ( P N (1)) N ≥ and the difference equation de-termine uniquely the sequence of polynomials ( P N ) N ≥ . To simplify the recurrence,we write Q N ( s ) = ( − N P N ( s −
1) and a N = Q N (0). The polynomials ( Q N ) satisfythe difference equations ∆ ω N +1 Q N +1 = Q N . We define the ω -descending factorial that form a triangular bases for the action ofthe operator ∆ ω on polynomials. Definition 6.15.
Let ω ∈ C ∗ . For s ∈ C and for an integer k ≥ , we define the ω -descending factorial as s [ k,ω ] = s ( s − ω ) . . . ( s − ( k − ω )For ω = 1 we get the usual descending factorial. A simple computation shows: Proposition 6.16.
We have ∆ ω s [ k +1 ,ω ] = ( k + 1) ωs [ k,ω ] Now we can give the general structure of the solutions ( Q N ). Proposition 6.17.
For N ≥ , we have Q N ( s ) = N X k =0 a N − k ω ω . . . ω k (cid:20) s [ k,ω N ] k ! + A N,k ( ω , . . . , ω N , s ) (cid:21) where the A N,k are polynomials in N + 1 variables and their total degree in the first N variables is strictly less than k . The coefficient a is an arbitrary integer. From this Proposition it is clear that the space of solutions Q N , and P N , is iso-morphic to Z N +1 by the one-to-one correspondence Q N ( a , a , . . . , a N ) ∈ Z N +1 .The proof of this Proposition follows by induction on N ≥
1, solving the differenceequation ∆ ω N +1 Q N +1 = Q N . For this, we develop the polynomials s [ k,ω N ] k ! + A N,k ( ω , . . . , ω N , s )in the bases ( s k ), then we change to the bases ( s [ ω N +1 ,k ] ) using the following Lemma: Lemma 6.18.
For n ≥ , s n = n X k =0 B n,k ( ω ) s [ ω,k ] where B n,n = 1 , B n,k ∈ Z [ X ] and deg B n,k ≤ n − k .Proof. We proceed by induction. The result is clear for n = 1, and developing s [ ω,n ] = s ( s − ω ) . . . ( s − ( n − ω ) we get s n = s [ ω,n ] − n X k =1 a k ω k s n − k and the induction hypothesis proves the result. (cid:3) Now we can study uniqueness conditions. A first result is a straightforward gener-alization by induction of the uniqueness result from Proposition 6.7.
Proposition 6.19.
Under the conditions as in Theorem 6.14, and if we assumethat for ≤ n ≤ N − , ω n +1 and ω n are Q -independent, then the hierarchy up to N ≥ , (Γ fn ( s | ω n )) ≤ n ≤ N is uniquely determined by its values (Γ fn ( k | ω n )) ≤ n ≤ N atsome integer k ≥ . If we assume some algebraic independence of the parameters, we have a muchstronger result.
N THE DEFINITION OF HIGHER GAMMA FUNCTIONS 29
Theorem 6.20.
Under the same conditions as in Theorem 6.14, and if we assumethat for ≤ n ≤ N , (18) [ Q [ ω , . . . , ω n ] : Q [ ω , . . . , ω n − ]] ≥ n + 1 then the hierarchy (Γ fn ( s | ω n )) ≤ n ≤ N is uniquely determined by any value Γ fN ( k | ω N ) at some integer point k ≥ .Proof. If we have two solutions ( g n ( s | ω n )) ≤ n ≤ N and (˜ g n ( s | ω n )) ≤ n ≤ N , the equality, g N ( k | ω N ) = ˜ g N ( k | ω N ), at the integer k ∈ C + , that is neither a zero nor pole of thefunctions, shows that the corresponding polynomials Q N and ˜ Q N satisfy Q N ( k ) − ˜ Q N ( k ) = a ∈ Z Then, using Proposition 6.17, this gives N X k =0 a N − k − ˜ a N − k ω ω . . . ω k (cid:20) s [ k,ω N ] k ! + A N,k ( ω , . . . , ω N , s ) (cid:21) = 0or, multiplying by ω ω . . . ω N , we get the algebraic relation( a − ˜ a ) s [ N,ω N ] N ! + . . . + ( a N − ˜ a N ) = 0where the dots are of degre < N in ω N . The degree assumption proves that a = ˜ a .Using the induction hypothesis on N (replacing f by g ( s | ω ) = ˜ g ( s | ω ), etc), we get a = ˜ a ,..., a N = ˜ a N . (cid:3) Using Proposition 6.17 we can give other uniqueness results and characterizations.To conclude this section, we note that Ruijsenaars (2000, [22]) exploited also thedifference equations and their minimal solutions to prove numerous properties ofBarnes multiple Gamma functions. Shintani (1976, [25]) extended Barnes approachto multiple Gamma functions to a several variable setting. Friedman and Ruijsenaars(2004, [6]) extended Shintani’smltiple Gamma functions. We can also apply our func-tional equation approach to define these several variables Gamma functions withoutBarnes-Hurwitz zeta functions and we will treat this case in a forthcoming article.
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