Multiplier sequences, classes of generalized Bessel functions and open problems
aa r X i v : . [ m a t h . C V ] A ug MULTIPLIER SEQUENCES, CLASSES OF GENERALIZED BESSEL FUNCTIONSAND OPEN PROBLEMS
GEORGE CSORDAS AND TAM ´AS FORG ´ACS
Abstract.
Motivated by the study of the distribution of zeros of generalized Bessel-type functions, theprincipal goal of this paper is to identify new research directions in the theory of multiplier sequences.The investigations focus on multiplier sequences interpolated by functions which are not entire and sums,averages and parametrized families of multiplier sequences. The main results include (i) the developmentof a ‘logarithmic’ multiplier sequence and (ii) several integral representations of a generalized Bessel-typefunction utilizing some ideas of G. H. Hardy and L. V. Ostrovskii. The explorations and analysis, augmentedthroughout the paper by a plethora of examples, led to a number of conjectures and intriguing open problems.MSC2000: Primary 30D10, 30D15, 33C20; Secondary 26C10, 30C15 Introduction
In 1905 G. H. Hardy [13] studied the following entire functions of exponential type, as generalizations of e z :(1.1) E s,a ( z ) := ∞ X n =0 ( n + a ) s z n n ! , s ∈ R , a ≥ . Although Hardy allowed the parameters to be complex numbers, in the present paper we will only considerparameters satisfying the restrictions in (1.1). Note that E ,a = e z , and for k ∈ N , E k,a = e z T k ( z ), where T k ( z ) is a polynomial of degree k . If a = 0, we set E s, = ∞ X n =1 n s x n n ! , s ∈ R . In [18], I. V. Ostrovskii describesthe real zeros of these generalized exponential functions. Theorem 1. ( [18, Theorem 2.5] ) Let E s,a be defined as in (1.1), and let k ∈ N . (a) For k < s < k + 1 , E s,a has only k + 1 real zeros. (b) For s < , E s,a does not have any real zeros. A consequence of Theorem 1 is that { ( k + a ) s } ∞ k =0 is not a multiplier sequence (cf. Definition 5) for non-integral or negative s . The observation that the sequence { /k ! } ∞ k =0 is a complex zero decreasing sequence(cf. Definition 4), however, motivates the study of functions of the form(1.2) B s,a ( z ) := ∞ X n =0 ( n + a ) s z n n ! n !along with the location of their zeros (see Section 3). We close this section with some definitions, and thegeneral question which led to most of the work and considerations in this paper. Definition 2.
A real entire function ϕ ( x ) = ∞ X k =0 γ k k ! x k is said to belong to the Laguerre-P´olya class , written ϕ ∈ L − P , if it admits the representation ϕ ( x ) = cx m e − ax + bx ω Y k =1 (cid:18) xx k (cid:19) e − x/x k , where b, c ∈ R , x k ∈ R \ { } , m is a non-negative integer, a ≥
0, 0 ≤ ω ≤ ∞ and ω X k =1 x k < + ∞ . efinition 3. A real entire function ϕ ( x ) = ∞ X k =0 γ k k ! x k is said to be of type I in the Laguerre-P´olya class,written ϕ ∈ L − P I , if ϕ ( x ) or ϕ ( − x ) admits the representation ϕ ( x ) = cx m e σx ω Y k =1 (cid:18) xx k (cid:19) , where c ∈ R , m is a non-negative integer, σ ≥ x k >
0, 0 ≤ ω ≤ ∞ and ω X k =1 x k < + ∞ . If γ k ≥ k = 0 , , . . . . , we write ϕ ∈ L − P + . Finally, L − P ( −∞ ,
0] denotes the class of functions in L − P whose zeros lie in ( −∞ , ϕ belongs to L − P I if and only if its Taylor coefficients are ofthe same sign, or alternate in sign. Thus L − P + ⊂ L − P I ⊂ L − P . Definition 4.
A sequence of real numbers { γ k } ∞ k =0 is called a complex zero decreasing sequence, or CZDS ,if the linear operator T defined by T [ x k ] = γ k x k has the property that for every real polynomial p ( x ), Z C ( T [ p ( x )]) ≤ Z C ( p ( x )) , where Z C ( p ) denotes the number of non-real zeros of the polynomial p , counting multiplicity. Definition 5.
A sequence of real numbers { γ k } ∞ k =0 is called a (classical ) multiplier sequence (of the firstkind), if the associated linear operator T defined by T [ x k ] = γ k x k , for k = 0 , , , . . . , has the property thatfor every real polynomial p ( x ), Z C ( T [ p ( x )]) = 0 whenever Z C ( p ( x )) = 0 . In the rest of the paper the term ‘multiplier sequence’ will refer exclusively to a classical multipliersequence. Also, by ‘applying a sequence to a function f ’, we simply mean the application of the operator T = { γ k } ∞ k =0 to f ; that is, if f ( x ) = P ∞ k =0 a k x k , then T [ f ( x )] := P ∞ k =0 γ k a k x k . The following is one of theessential results concerning the characterization of multiplier sequences, due to P´olya and Schur. Theorem 6. ( [19] or [17, Ch. II.] ) Let { γ k } ∞ k =0 be a sequence of real numbers. The following are equivalent: (i) { γ k } ∞ k =0 is a multiplier sequence; (ii) (Algebraic characterization) for each n ∈ N ; n X k =0 (cid:18) nk (cid:19) γ k x k ∈ L − P I ;(iii) (Transcendental characterization) ∞ X k =0 γ k k ! x k ∈ L − P I. Definition 7.
Let T = { γ k } ∞ k =0 be a sequence of real numbers. For n ∈ N , we definite the n th Jensenpolynomial associated with the sequence T to be g n ( x ) := T [(1 + x ) n ] = n X k =0 (cid:18) nk (cid:19) γ k x k . Given Theorem 6, a reasonably easy way to show that a sequence T = { γ k } ∞ k =0 is not a multiplier sequenceis to demonstrate the existence of a Jensen polynomial associated with T possessing non-real zeros.The following problem motivated most of the investigations in the present paper. The original nomenclature for such sequences did not include the adjective classical . Indeed, P´olya and Schur in [19] calledthese simply
Faktorenfolgen erster Art . More recently, research has focused on sequences giving rise to linear operators that arediagonal with respect to a basis other than the standard one, necessitating the introduction of modifiers. We now talk aboutHermite-, Laguerre-, Legendre- and Chebyshev-multiplier sequences (see for example [4], [11], [5], [12], [10], [23]). Consequently,we use the word ‘classical’ to describe multiplier sequences whose operators are diagonal with respect to the standard basis. roblem 8. Characterize all non-negative sequences { γ k } ∞ k =0 such that if f ( x ) = ∞ X k =0 γ k k ! x k is an entire function, then(1.3) F p ( x ) = ∞ X k =0 γ k k !Γ( k + p + 1) x k ∈ L − P , for − p / ∈ N . We regard the function F p ( x ) in (1.3) as a generalized Bessel-type function. In support of this view, werecall that the modified Bessel function of the first kind of order p ([2, p. 228] or [20, p. 116]) is defined as(1.4) I p ( x ) := (cid:16) x (cid:17) p ∞ X k =0 ( x/ k k !Γ( k + p + 1) = ( x/ p Γ(1 + p ) F (cid:18) − ; 1 + p ; x (cid:19) , ( − p / ∈ N ) , where F ( − ; b ; x ) := ∞ X k =0 x k ( b ) k k ! is the hypergeometric function, and ( b ) p := b ( b + 1) · · · ( b + n −
1) = Γ( b + n )Γ( b )is the rising factorial, − b / ∈ N , n ∈ N and ( b ) = 1. Simple transformations show that with γ k = 1 / k for k = 0 , , , . . . , the function F ( x ) in (1.3) reduces to the modified Bessel function I ( √ x ). We emphasizehere that our generalizations of the Bessel functions are different from those appearing in the literature.Indeed, see for example ´A. Baricz’ excellent monograph [3], where he studies, for suitable parameters b and c , the function w p ( z ) = ∞ X n =0 ( − c ) n n !Γ( p + n + ( b + 1) / (cid:16) z (cid:17) n + p , and refers to it as the generalized Bessel function of the first kind of order p .In reference to Problem 8, it is clear that F p ( x ) ∈ L − P whenever { γ k } ∞ k =0 is a multiplier sequence.Thus, the task is to characterize non-negative real sequences { α k } ∞ k =0 , which are not multiplier sequences,but for which the ‘composed’ sequence { α k /k ! } ∞ k =0 is a multiplier sequence. Canonical examples appear tobe difficult to construct. As an illustrative example, the sequence (cid:8) k + 2 (cid:9) ∞ k =0 is not a multiplier sequence,since ∞ X k =0 k + 2 k ! x k = e x (2 + x + x ) / ∈ L − P (cf. (iii), Theorem 6). On the other hand, one can readilycheck that F ( x ) = ∞ X k =0 k + 2 k ! k ! x k = (2 + x ) I ( √ x ) = F ( − ; 1; x ) ∈ L − P , and whence n k +2 k ! o ∞ k =0 is a multiplier sequence (see Proposition 35).The rest of the paper is organized as follows. In connection with Problem 8, Section 2 investigates alogarithmically interpolated sequence (Theorem 11 and Corollary 13), and sums and averages of multipliersequences (Theorem 14 and Corollary 16). By adopting some of the ideas of Hardy [13] and Ostrovskii [18],the main results of Section 3 furnish several integral representations of the entire function f ( x ) = P ∞ k =0 √ kk ! k ! x k (cf. Theorem 22). Motivated by the work in Section 3 (see, in particular, Example 20), Section 4 providesgenerating functions which yield new families of multiplier sequences varying smoothly with a parameter. Thegoal of Section 5 is multifold: (i) to indicate possible applications of the foregoing results in the theory Besselfunctions or hypergeometric functions, (ii) to highlight additional propositions (see, for example, Proposition35 ) supporting the conjecture in Section 3.1 and (iii) to cite additional examples and list problems whicharose during the analysis of various sequences, but remain unsolved at this time.2. The log sequence
The function B s,a ( z ) in equation (1.2) can be regarded as a ‘generalized’ exponential function ´a la Hardy,whose Taylor coefficients are interpolated by the function g ( x ) = ( x + a ) s Γ( x + 1) , s ∈ R , a ≥
0. The restriction on a assures that for s ∈ N , g ( x ) interpolates a multiplier sequence. Notice that for non-integral s > g ( x ) s not entire. Thus, our explorations differ from the traditional approach, where the interpolating functionis almost exclusively taken to be entire. We first look at a logarithmically interpolated sequence and thefollowing real entire function: f ( x ) = ∞ X k =0 ln( k + 2) k ! k ! x k . When understood as an alteration of the modified Bessel function of the first kind of order zero (see (1.4)) I ( x ) = ∞ X k =0 ( x/ k k ! k ! , one would attempt to establish the reality of zeros of f by showing that { ln( k + 2) } ∞ k =0 is a multipliersequence. This is not the case, however. If T := { ln( k + 2) } ∞ k =0 , then the zeros of the Jensen polynomial g ( x ) = T [(1 + x ) ] = ln 2 + 3 x ln 3 + 3 x ln 4 + x ln 5are x = − . . . . and x , = − . · · · ± i . . . . . We are thus led to consider the newsequence T := { ln ( k + 2) /k ! } ∞ k =0 and the associated entire function: f ( x ) := T [ e x ] = T " ∞ X k =0 x k k ! = ∞ X k =0 (cid:18) ln( k + 2) k ! (cid:19) x k k ! . While we believe that the sequence { ln( k + 2) /k ! } ∞ k =0 is a multiplier sequence, we were able to establishsuch a claim only for an approximating sequence. In order to be able to formulate our theorem (cf. Theorem11), we need a few preliminary results. Recall (see [21, p. 8]) thatlim n →∞ ( H n − ln( n )) = γ, where H n := n X k =1 k is the n th harmonic number, and γ is the Euler-Mascheroni constant. Thus, for n ≫ H n +2 − γ ≈ ln( n + 2) , and consequently, H n +2 − γn ! ≈ ln( n + 2) n ! . Proposition 9. If n ∈ N , then H n = n X k =1 (cid:18) nk (cid:19) ( − k − k . Proof.
The proof is based on an induction argument, treating the even and odd cases separately. (cid:3)
The following result is well known, but for the sake of completeness, we include a short proof of it here.
Proposition 10. ( [9, vol. I, p. 15] ). Let ψ ( x ) denote the digamma function ψ ( x ) := ddx ln Γ( x ) = Γ ′ ( x )Γ( x ) . Then ∞ X n =0 H n +2 n ! x n n ! = ∞ X n =0 γ + ψ (3 + n ) n ! x n n ! . Proof.
It is well known (see [21, pp. 11-12]) thatΓ ′ (1) = − γ, andΓ( x + 1) = x Γ( x ) , for all x > . It follows that ψ ( x + 1) = ddx ln Γ( x + 1) = 1 x + ddx ln Γ( x ) = 1 x + ψ ( x ) . tarting with ψ (2) = 1 + ψ (1) = H − γ , a simple inductive argument establishes that ψ ( n + 1) = H n − γ for all n ∈ N , which completes the proof. (cid:3) Theorem 11.
The entire function f ( x ) := ∞ X n =0 ( H n +2 − γ ) n ! x n n ! belongs to L − P + , and hence the sequence n ( H n +2 − γ ) n ! o ∞ n =0 is a multiplier sequence. Remark 12.
Before proving Theorem 11, we observe that the class of functions L − P ( −∞ ,
0] (seeDefinition 3) is not closed under differentiation. For example, ϕ ( x ) := e − x + x ( x + 1) ∈ L − P ( −∞ , ϕ ′ ( x ) = e − x + x (2 − x − x ) has a positive zero. Proof.
We note that H n − γ > n ∈ N (see [21, p. 9]), and hence the Taylor coefficients of f are allpositive. With the aid of Proposition 10, we can express f ( x ) as(2.1) f ( x ) = ∞ X n =0 ψ ( n + 3) n ! x n n ! . Since 1Γ( x ) ∈ L − P ( −∞ ,
0] (cf. Definition 3), it follows that (cid:18) x ) (cid:19) ′ = − Γ ′ ( x )Γ ( x ) = − ψ ( x )Γ( x ) ∈ L − P . It is known that for x >
0, the only extremum of Γ( x ) occurs at x = 1 . . . . (see, for example, [2, p. 90]).Since x corresponds to a minimum of Γ( x ), we infer that Γ ′ ( x ) and ψ ( x ) are both negative on the interval(0 , x ) and are both positive on ( x , ∞ ). It now follows that all the zeros of the entire function ϕ ( x ) := ψ ( x )Γ( x ) lie in ( −∞ , ϕ ( x + 3) ∈ L − P ( −∞ ,
0) and consequently, by Laguerre’s theorem ([7, Theorem4.1(3)]), the sequence T := { ϕ ( k + 3) } ∞ k =0 is a CZDS (cf. Definition 4) and a fortiori T is a multipliersequence. Thus T ( e x ) = ∞ X k =0 ψ ( k + 3)Γ( k + 3) x k k ! = ∞ X k =0 ψ ( k + 3)( k + 2)! x k k ! ∈ L − P + . Finally, applying the multiplier sequence { ( k + 2)( k + 1) } ∞ k =0 to T ( e x ) yields the desired result (cf. (2.1)): ∞ X k =0 ( k + 2)( k + 1) ψ ( k + 3)( k + 2)! x k k ! = ∞ X k =0 ψ ( k + 3) k ! x k k ! = f ( x ) ∈ L − P + . (cid:3) Corollary 13.
For t ∈ R , let { t } = t − ⌊ t ⌋ , where ⌊ t ⌋ denotes the greatest integer less than or equal to t .Then the sequence ln( k + 2) + Z ∞ k +2 { t } t dtk ! ∞ k =0 is a multiplier sequence.Proof. In [15, p. 540] J. Lagarias states that for all k ≥ H k = ln k + γ + Z ∞ k { t } t dt. Rearranging equation (2.2) and applying Theorem 11 yields the result. (cid:3) .1. Sums and averages.
The harmonic approximation to the logarithm motivates the study of sumsand averages of initial segments of multiplier sequences (and sequences in general), and whether or notsuch derived sequences are again multiplier sequences. We begin by noting that if { γ k } ∞ k =0 is a multipliersequence, then the sequences(2.3) T = k X j =0 γ j ∞ k =0 and T = k + 1 k X j =0 γ k ∞ k =0 need not be multiplier sequences. Indeed, if { γ k } ∞ k =0 = { /k ! } ∞ k =0 , then T [(1 + x ) ] = 1 + 8 x + 15 x + 323 x + 6424 x / ∈ L − P ,T [(1 + x ) ] = 1 + 3 x + 52 x + 23 x / ∈ L − P . The converse implication however is true, if the sequence { γ k } ∞ k =0 can be interpolated by a polynomial withnon-negative coefficients. Theorem 14.
For k ∈ N let γ k = p ( k ) , where p ( x ) := m X j =0 a j x j , and a j ≥ . Set S ( k ) = k X j =0 γ j , and A ( k ) = γ + γ + · · · + γ k k + 1 , k ≥ . If the average sequence { A ( k ) } ∞ k =0 is a multiplier sequence, then so is the sequence { γ k } ∞ k =0 .Proof. We shall arrive at the desired result by demonstrating that the function f ( x ) = ∞ X k =0 p ( k ) x k k !belongs to L − P + . To this end consider Q ( x ) = e − x ∞ X n =0 A ( n ) x n n != e − x ∞ X n =0 n X k =0 p ( k ) ! n + 1 x n n ! ∈ L − P , here the membership in L − P follows, since by assumption, { A ( k ) } ∞ k =0 is a multiplier sequence. Conse-quently, xQ ( x ) and its derivative both belong to L − P . We now calculate D [ xQ ( x )] = D " e − x ∞ X n =0 n X k =0 p ( k ) ! n + 1 x n +1 n ! D := ddx (cid:19) = e − x " ∞ X n =0 n X k =0 p ( k ) ! x n n ! − ∞ X n =0 n X k =0 p ( k ) ! x n +1 ( n + 1)! = e − x " p (0) + ∞ X n =0 n +1 X k =0 p ( k ) ! x n +1 ( n + 1)! − ∞ X n =0 n X k =0 p ( k ) ! x n +1 ( n + 1)! = e − x " p (0) + ∞ X n =0 p ( n + 1) x n +1 ( n + 1)! = e − x " ∞ X n =0 p ( n ) x n n ! = e − x f ( x ) . Thus e x D [ xQ ( x )] = f ( x ) ∈ L − P . The assumption that a j ≥ j ∈ N ensures that in fact f ∈ L − P + , and our proof is complete. (cid:3) We offer two corollaries of Theorem 14.
Corollary 15. If { A ( k ) } ∞ k =0 is a multiplier sequence, then so is { S ( k ) } ∞ k =0 .Proof. The result is immediate, since { ( k + 1) } ∞ k =0 is a multiplier sequence. (cid:3) Corollary 16.
Suppose that p is as in the statement of Theorem 14, and let m = deg p . If ≤ ℓ ≤ m + 1 and the sequence (cid:26) S ( k )( k + 1) ℓ (cid:27) ∞ k =0 is a multiplier sequence, then so is the sequence { , , , . . . , p (0) |{z} ℓ th slot , p (1) , p (2) , . . . } . Proof.
The proof is essentially the same as that of Theorem 14, if one differentiates x ℓ e Q ( x ), where e Q ( x ) isan appropriately modified version of Q ( x ). In particular, D " e − x x ℓ ∞ X k =0 S ( k )( k + 1) ℓ x k k ! ! = D " e − x ∞ X k =0 S ( k ) x k + ℓ ( k + ℓ )! ! = e − x " ∞ X k = ℓ − p ( k − ℓ + 1) x k k ! ∈ L − P , and hence the sequence { , , , . . . , p (0) , p (1) , p (2) , . . . } is a multiplier sequence. (cid:3) We remark that Corollary 16 gives a sufficient condition when one can pre-concatenate a polynomiallyinterpolated multiplier sequence with a string of zeros and thus obtain another multiplier sequence.
Example 17.
This example is an illustration of Corollary 16 under the assumption that p ( x ) = m Y j =1 ( x + j ) ∈ L − P + , m ≥ . or such polynomials we claim, that if S ( n ) := P mk =0 p ( k ), then(2.4) S ( n ) = 1 m + 1 m +1 Y k =1 ( k + n ) for all n ∈ N . Proof of Claim.
We proceed by double induction. Fix the degree m of p , and let n = 1. S (1) = X k =0 p ( k ) = m ! + m Y k =1 ( k + 1) = m !( m + 2) = 1 m + 1 m +1 Y k =1 ( k + 1) . Suppose now that equation (2.4) holds for some n ≥
1. Then S ( n + 1) = p ( n + 1) + S ( n )= m Y k =1 ( n + 1 + k ) + 1 m + 1 m +1 Y k =1 ( k + n )= 1 m + n + 2 m +1 Y k =1 ( k + n + 1) + n + 1 m + 1 1 m + n + 2 m +1 Y k =1 ( k + n + 1)= 1 m + 1 m +1 Y k =1 ( k + n + 1) . Since m was arbitrary, the claim follows. (cid:3) From equation (2.4) one can readily deduce that the sequence (cid:26) S ( k )( k + 1) ℓ (cid:27) ∞ k =0 is a multiplier sequence forany 1 ≤ ℓ ≤ m + 1. Corollary 16 implies that the sequences { p m (0) , p m (1) , p m (2) , . . . }{ , p m (0) , p m (1) , p m (2) , . . . }{ , , p m (0) , p m (1) , p m (2) , . . . } ... { , , . . . , p m (0) | {z } m +1st slot , p m (1) , . . . } are all multiplier sequences. By way of illustration, if m = 4 and ℓ = 2, one obtains the multiplier sequence { , , , , , , . . . , ( k + 2)!( k − , . . . } , and thus, the entire function ∞ X k =2 ( k + 2)!( k − x k k ! = e x x ( x + 2)( x + 6) ∈ L − P + . Example 18.
The converse of Theorem 14 is false in general. That is, if { p ( k ) } ∞ k =0 is a multiplier sequence,the average { A ( k ) } ∞ k =0 need not be a multiplier sequence. The sequence { p ( k ) } ∞ k =0 = (cid:8) k + k (cid:9) ∞ k =0 is amultiplier sequence, since ∞ X k =0 k + k k ! x k = e x (1 + x ) ∈ L − P + . The average sequence { A ( k ) } ∞ k =0 = (cid:26)
13 (3 + 2 k + k ) (cid:27) ∞ k =0 however is not a multiplier sequence, because ∞ X n =0 n + n x n n ! = 13 e x (3 + 3 x + x ) / ∈ L − P . s can be verified by the reader, remarkably, both the sequence (cid:8) (1 + k + k ) (cid:9) ∞ k =0 and its average (cid:26) k + 386 k + 384 k + 246 k + 90 k + 15 k ) (cid:27) ∞ k =0 are multiplier sequences.One may wonder whether requiring p to belong to L − P + could result in a partial converse of Theorem14. This is not the case. Setting p ( x ) = ( x + 4) , we see that S ( k ) = (1 + k )(96 + 25 k + 2 k ), and ∞ X k =0 S ( k ) k ! x k = e x x + 33 x + 2 x ) / ∈ L − P . n ( k + a ) s k ! o ∞ k =0 type sequences In general, the sequence (cid:26) k + a ) k ! (cid:27) ∞ k =0 is not a multiplier sequence. For example, if a = 1 /
2, then theJensen polynomial (cf. Definition 7) g ( x ) = X k =0 (cid:18) k (cid:19) x k ( k + 1 / k ! = 2 + 83 x + 65 x + 421 x + 1108 x has two non-real zeros. Lemma 19. ( [6, Proposition 40] ) The sequence (cid:26) k + a ) k ! (cid:27) ∞ k =0 is a multiplier sequence for every a ∈ N .Proof. The result follows immediately from the fact that for any a ∈ N ,( k + 1)( k + 2) · · · ( k + a − k + a + 1) = ( k + 1)( k + 2) · · · ( k + a − · · · · k ( k + 1)( k + 2) · · · ( k + a )= 1 k !( k + a ) . (cid:3) The sequence n √ k/k ! o ∞ k =0 . We conjecture that the function ϕ ( x ) = ∞ X k =0 √ kk ! x k k ! belongs to L − P + .Our contention is that among the sequences of the form { k s /k ! } ∞ k =0 with s non-integral, the case s = 1 / Example 20.
The sequence (cid:8) k / /k ! (cid:9) ∞ k =0 is not a multiplier sequence. In fact, if we consider the Jensenpolynomials associated with f ( x ) = ∞ X k =0 k / k ! x k we find that the sixth Jensen polynomial g ( x ) = 6 x + 15 x / + 10 x / + 5 x · / + x / + x · / has has only four real zeros, along with a pair of non-real zeros. This phenomenon persists for small s , butappears to change when s = 1 /
2. In this case all the Jensen polynomials that we tested have only real zeros.There is a marked sparsity of known multiplier sequences which involve √ k non-trivially. We offer herethe following examples.(i) The sequence n cosh( √ k ) o ∞ k =0 is a multiplier sequence. This follows from the containmentcosh √ x = ∞ X k =0 x k (2 k )! = ∞ X k =0 k !(2 k )! x k k ! = ∞ Y k =0 x (cid:0) πk + π (cid:1) ! ∈ L − P + , together with Laguerre’s theorem ([7, Theorem 4.1(3)]). ii) In order to formulate the second example, we first recall that a sequence { γ k } ∞ k =0 of non-negativereal numbers is said to be rapidly decreasing , if γ k ≥ γ k − γ k +1 for all k ∈ N ([7, p.438]). Suchsequences are known to be multiplier sequences. We now note that if { γ k } ∞ k =0 is rapidly decreasing,then so is n √ kγ k o ∞ k =0 , which in turn makes the latter also a multiplier sequence.It is not known whether ϕ ( x ) = ∞ X k =0 √ kk ! x k k ! belongs to L − P + . The entire function ϕ ( x ) is however Hurwitzstable; that is, all of its zeros lie in the closed left half-plane. Lemma 21.
The entire function ϕ ( x ) = ∞ X k =0 √ kk ! x k k ! is Hurwitz stable.Proof. The work of Ostrovskii ([18, Corollary 2.2]) shows that the entire function f ( x ) = ∞ X k =0 √ kk ! x k is Hurwitz stable. Thus, f ( − ix ) has all of its zeros in the closed upper half-plane, and hence by theHermite-Biehler theorem, f ( − ix ) = p ( x ) + iq ( x ), where p ( x ) and q ( x ) have only real, interlacing zeros. Let T = { /k ! } ∞ k =0 . Then the entire function T [ f ( − ix )] = T [ p ( x )] + iT [ q ( x )] also has all its zeros in the upperhalf-plane [16, p. 342]. Finally, the change of variables x ix shows that ϕ ( x ) is Hurwitz stable. (cid:3) We close this section by giving two integral representations for the function ϕ ( x ) = ∞ X k =0 √ kk ! x k k ! , which mayhelp in determining whether it belongs to L − P + . We arrive at the first of the two representations byadapting the main ideas in [18, Section 3]. Theorem 22. If f ( x, t ) := ∞ X n =0 x n ( e − t ) n n ! n ! , then (3.1) ∞ X n =0 √ nn ! x n n ! = − √ π Z ∞ [ f ( x, u ) − f ( x, duu / . Proof.
We start with the following generalizations of the modified Bessel function: B (0 , x ) := ∞ X n =0 x n n ! n ! ,B ( s, x ) := ∞ X n =0 n s n ! x n n ! ( s > . (3.2)Differentiating (3.2) with respect to x yields ddx B ( s, x ) = ∞ X n =1 n s +1 n ! x n − n ! , and whence x ddx [ B ( s, x )] = B ( s + 1 , x ) , s ≥ . Now set f ( x, t ) := ∞ X n =0 x n ( e − t ) n n ! n ! , and consider the generating relation(3.3) f ( x, t ) = ∞ X n =0 ( − n Q n ( x ) t n n ! . ifferentiating (3.3) with respect to t and x yields ∞ X n =0 − nx n ( e − t ) n n ! n ! = ∞ X n =0 ( − n +1 Q n +1 ( x ) t n n ! , and(3.4) ∞ X n =0 nx n − ( e − t ) n n ! n ! = ∞ X n =0 ( − n Q ′ n ( x ) t n n ! . (3.5)By equating the coefficients of t n in (3.4) and (3.5), we deduce that Q n +1 ( x ) = xQ ′ n ( x ) , n ∈ N . Notice that Q ( x ) = B (0 , x ), and since the sequences { B ( n, x ) } and { Q n ( x ) } satisfy the same recurrencerelation, we conclude that Q n ( x ) = B ( n, x ) , n ∈ N . With the aid of (3.3) we now give an integral representation for B ( s, x ) for non-integral values of s . For k < s < k + 1, k ∈ N , the Cauchy-Saalsch¨utz formula ([22, Sec. 12.21]) yieldsΓ( − s ) = Z ∞ e − t − k X j =0 ( − j t j j ! dtt s +1 . The change of variables t = nu gives n s = 1Γ( − s ) Z ∞ e − nu − k X j =0 ( − j ( nu ) j j ! duu s +1 . Consequently, B ( s, x ) = ∞ X n =1 n s n ! x n n !(3.6) = ∞ X n =1 x n n ! n ! 1Γ( − s ) Z ∞ e − nu − k X j =0 ( − j ( nu ) j j ! duu s +1 = 1Γ( − s ) Z ∞ f ( x, u ) − k X j =0 ( − j Q j ( x ) j ! u j duu s +1 , ( k < s < k + 1 , k ∈ N ) . Thus, setting k = 0 and s = 1 / (cid:3) Corollary 23.
The following representation is valid: (3.7) ∞ X n =0 √ nn ! x n n ! = 12 √ π Z [ B (0 , x ) − B (0 , xv )] dvv ( − ln( v )) / Proof.
The above representation follows directly from (3.1) with the change of variables v = e − u . We remarkthat the convergence of the integral for every x ∈ C in the statement of the corollary can be directly verifiedusing the identity Z − v n v ( − ln( v )) s dv = − n s Γ( − s ) , n ∈ N , < s < . (cid:3) We conclude this section with two more integral representations which could be of use in further investi-gations. With I p denoting the modified Bessel functions (cf. (1.4)), the following formulæ ϕ ( x ) := ∞ X n =0 √ nn ! x n n ! = 1 √ π Z √ xI (2 √ xt ) √ t √− ln t dt, nd ϕ ′ ( x ) = 1 √ π Z I (2 √ xt ) √− ln t dt. Transformations of multiplier sequences
This section is motivated by the observation that in some sense the sequence { (ln( k + 2)) /k ! } ∞ k =0 ‘liesbetween’ two multiplier sequences (namely { } ∞ k =0 and { /k ! } ∞ k =0 ), since k ! ≻ ln( k +2) ≻ { ( H k +2 − γ ) /k ! } ∞ k =0 is a multiplier sequence, which itself is an approximation tothe sequence { (ln( k + 2)) /k ! } ∞ k =0 . Thus the notions of deformation, perturbation and general transformationof multiplier sequences arise naturally. A quite fruitful way of obtaining new multiplier sequences is to identifythose transformations, which when applied to multiplier sequences, result again in multiplier sequences. Weformulate the following general problem. Problem 24.
Let { α k } ∞ k =0 and { β k } ∞ k =0 be multiplier sequences of non-negative real numbers. Characterizeall functions Ψ : R × R → R such that { Ψ( α k , β k ) } ∞ k =0 is again a multiplier sequence.There are some simple functions with this property, such as the projection onto either coordinate axis, orthe function Ψ( x, y ) = xy . Lemma 25.
Convex combinations of two multiplier sequences need not be multiplier sequences. That is, ingeneral Ψ λ ( α k , β k ) := λα k + (1 − λ ) β k ( λ ∈ [0 , ) is not a solution to Problem 24.Proof. Let { α k } ∞ k =0 = (cid:8) k + k + 1 (cid:9) ∞ k =0 and { β k } ∞ k =0 = (cid:8) k ! (cid:9) ∞ k =0 . With λ = , we get the sequence n k + k +110 + k ! o ∞ k =0 , which, when applied to (1 + x ) yields the polynomial p ( x ) = 1 + 245 x + 6910 x + 295 x + 17180 x / ∈ L − P . (cid:3) Simple examples show that there exists multiplier sequences of non-negative terms, whose linear combi-nation is again a multiplier sequence.
Proposition 26. ( [20, p. 198] ) Suppose that p, q ∈ L − P + are polynomials with strictly interlacing zeros.Then for any a, b ∈ R + , the polynomial aq ( x ) + bp ( x ) ∈ L − P + . Thus, any positive linear combination ofthe multiplier sequences { q ( k ) } ∞ k =0 and { p ( k ) } ∞ k =0 is again a multiplier sequence. Lemma 27.
Convex geometric combinations of two multiplier sequences need not be multiplier sequences.That is, in general Ψ λ ( α k , β k ) := α λk β (1 − λ ) k , λ ∈ [0 , , is not a solution to Problem 24.Proof. Consider { α k } ∞ k =0 = (cid:8) k + k + 1 (cid:9) ∞ k =0 and { β k } ∞ k =0 = { } ∞ k =0 . Calculating Ψ / ( α k , β k )[(1+ x ) ] givesthe polynomial q ( x ) = 1 + 4 √ x + 6 √ x + 4 √ x + √ x , which has two non-real zeros. (cid:3) The following proposition provides a ‘continuously deformed’ family of multiplier sequences.
Proposition 28.
Suppose that ϕ ( x ) = ∞ X k =0 γ k k ! x k ∈ L − P + . Then for t ∈ [0 , the sequence { B ϕk ( t ) } ∞ k =0 is a multiplier sequence, where (4.1) B ϕk ( t ) = k X j =0 (cid:18) kj (cid:19) (1 − t ) j γ k − j t k − j . roof. This is a straightforward consequence of the generating relation e (1 − t ) x ϕ ( xt ) = ∞ X k =0 B ϕk ( t ) k ! x k . (cid:3) Note that { B ϕk (0) } ∞ k =0 = { γ } ∞ k =0 and { B ϕk (1) } ∞ k =0 = { γ k } ∞ k =0 . Corollary 29.
Let ϕ, Φ ∈ L − P + , and let { B ϕk ( t ) } ∞ k =0 and (cid:8) B Φ k ( s ) (cid:9) ∞ k =0 be the associated multipliersequences (cf. equation (4.1)). Then the sequence (4.2) n C ϕ, Φ k ( t, s ) o ∞ k =0 := k X j =0 (cid:18) kj (cid:19) B ϕj ( t ) B Φ k − j ( s ) ∞ k =0 is a multiplier sequence for all ( t, s ) ∈ [0 , × [0 , .Proof. The representation (4.2) is a consequence of the generating relation(4.3) e ((1 − t )+(1 − s )) x ϕ ( xt )Φ( xs ) = ∞ X k =0 C ϕ, Φ k ( t, s ) k ! x k . (cid:3) Remark 30.
Several observations are in order.(a) If f : [0 , → [0 , { C ϕ, Φ k ( t, f ( t )) } ∞ k =0 is a multiplier sequence for all t ∈ [0 , ϕ ( t ) = ∞ X k =0 γ k k ! x k ∈ L − P + and Φ( t ) = ∞ X k =0 β k k ! x k ∈ L − P + . Then the Cauchy product of ϕ ( t ) and Φ( t ) is also in L − P + ; that is, ϕ ( t ) Φ( t ) = ∞ X k =0 c k t k k ! ∈ L − P + , w here c k := k X j =0 (cid:18) kj (cid:19) γ j β k − j , and hence { c k } ∞ k =0 is a multiplier sequence.(c) The polynomial B ϕk ( t ) (see (4.1)) can also be expressed in terms of the Jensen polynomials { g j ( t ) } ∞ j =0 associated with ϕ ( t ): B ϕk ( t ) = k X j =0 (cid:18) kj (cid:19) g j ( t )( − j + k t k − j . (d) Finally, we remark that for each fixed k ∈ N , the polynomial B ϕk ( t ) has only real zeros. A shortproof of this assertion is as follows. For fixed k ∈ N and t = 0, a calculation shows that t k B ϕk (cid:18) t (cid:19) = t k k X j =0 (cid:18) kj (cid:19) (cid:18) − t (cid:19) j γ k − j t k − j = k X j =0 (cid:18) kj (cid:19) ( t − j γ k − j , and whence we infer that B ϕk ( t ) has only real zeros. Here a caveat is in order since, in general, thezeros of B ϕk ( t ) need not be all negative. Definition 31.
Let γ k ∈ R for k = 0 , , , . . . . We say that the sequence { γ k } ∞ k =0 has a C k -representation,if there exist functions ϕ, Φ ∈ L − P + (not necessarily distinct) and s, t ∈ R such that γ k = C ϕ, Φ k ( t, s ) forall k ∈ N , where C ϕ, Φ k ( t, s ) is as defined in equation (4.2). Theorem 32. (1) Every polynomially interpolated multiplier sequence of non-negative terms can be writtenas a sequence n C ϕ, Φ k ( t, s ) o ∞ k =0 for some functions ϕ, Φ ∈ L − P + and ( t, s ) ∈ [0 , × [0 , . The choice of ϕ, Φ , t and s in this representation need not be unique.(2) Every geometric multiplier sequence (i.e., a sequence of the form (cid:8) r k (cid:9) ∞ k =0 , r ∈ R ) has a C k -representation. roof. (1) For any p ∈ R [ x ], ∞ X k =0 p ( k ) k ! x k = g p ( x ) e x , where g p ( x ) = a + n X j =1 n X k = j a k S ( k, j ) x j ∈ L − P + , and S ( k, j ) denote Stirling numbers of the second kind (see, for example, [14, Ch.7]). Suppose now that { γ k } ∞ k =0 is a multiplier sequence of non-negative terms, and suppose that p ∈ R [ x ] is such that p ( k ) = γ k for k = 0 , , , . . . . The equation(4.4) e ((1 − t )+(1 − s )) x ϕ ( xt )Φ( xs ) = g p ( x ) e x leads to the identification { γ k } ∞ k =0 = n C ϕ, Φ k ( t, s ) o ∞ k =0 . Indeed, equation (4.4) will be satisfied if (a) 1 = t + s ,and (b) ϕ ( xt )Φ( xs ) = g p ( x ). The selection t = 1 , s = 0, Φ( x ) ≡ ϕ ( x ) = g p ( x ) obviously satisfies ( a )and ( b ). Thus, { γ k } ∞ k =0 = (cid:26) C g p ( x ) , k (1 , (cid:27) ∞ k =0 . The non-uniqueness is easy to ascertain, for any g p ( x ) ∈ L − P + of degree two or higher has distinct factorizations into products of the form g p ( x ) = q ( x ) q ( x ),with q ( x ) , q ( x ) ∈ L − P + .(2) Again, solving (4 .
4) with e rx on the right hand side is possible with ϕ ≡ Φ ≡ t + s = 2 − r .In particular, (cid:8) r k (cid:9) ∞ k =0 = n C , k ( t, − t − r ) o ∞ k =0 . We remark that the restrictions on t, s force r ∈ [0 , ϕ and Φ. (cid:3) Example 33.
Suppose that ϕ ( x ) = Φ( x ). In this case we have C ϕ,ϕk ( t, s ) = k X j =0 (cid:18) kj (cid:19) B ϕj ( t ) B ϕk − j ( s ) , s, t ∈ R . For t, s ∈ (0 , c = t − t , c = s − s , and c = 1 − t − s . Then(4.5) C ϕ,ϕk ( t, s ) = (1 − s ) k k X j =0 (cid:18) kj (cid:19) c j g j ( c ) g k − j ( c ) , where g k ( x ) denotes the k th Jensen polynomial associated with the real entire function ϕ . In particular, if t = s = , then C ϕ,ϕk (cid:18) , (cid:19) = (cid:18) (cid:19) k k X j =0 (cid:18) kj (cid:19) g j (1) g k − j (1) . For the readers convenience, we give here (without proof) a teaser of choices for ϕ in equation (4.5), and theresulting family of multiplier sequences.(1) If we set ϕ ( x ) = ∞ X k =0 x k k ! k ! , and select s, t ∈ (0 , { C ϕ,ϕk ( t, s ) } ∞ k =0 = (1 − s ) k k X j =0 (cid:18) − t − s (cid:19) j (cid:18) kj (cid:19) L j (cid:18) tt − (cid:19) L k − j (cid:18) ss − (cid:19) ∞ k =0 , where L j ( x ) denotes the j th Laguerre polynomial (see for example [21, Ch. 12]).
2) The choice ϕ ( x ) = ∞ X k =0 x k (2 k )! , and s, t ∈ (0 ,
1) leads to the following multiplier sequences involvinghypergeometric functions: { C ϕ,ϕk ( t, s ) } ∞ k =0 = (1 − s ) k k X j =0 (cid:18) kj (cid:19) (cid:18) − t − s (cid:19) j F (cid:20) − j ; 12 ; t t − (cid:21) F (cid:20) − ( k − j ); 12 ; s s − (cid:21) ∞ k =0 . (3) Finally, selecting ϕ ( x ) = e rx , and s, t ∈ (0 ,
1) leads to the multiplier sequences { C ϕ,ϕk ( t, s ) } ∞ k =0 = (cid:8) (2 + ( s + t )( r − k (cid:9) ∞ k =0 . Scholia and Open problems
In the applications of the theory of Bessel functions (or hypergeometric functions p F q ) it is frequently im-portant to determine the distribution of zeros of certain combinations of Bessel functions (or hypergeometricfunctions). In this section, we propose some techniques involving multiplier sequences that may shed lighton several questions and intriguing problems that arose in the course of our analysis, but remain unsolvedat this time. Generalities aside, we commence here with a concrete example to illustrate the ideas involved. Example 34.
Let p ( x ) := cx + ax + b , where a, b, c ≥
0. If a = b = 1 and c >
0, so that p ( x ) = cx + x + 1,then ∞ X k =0 p ( k ) k ! x k = e x (1 + x )(1 + cx ) ∈ L − P + , and whence { p ( k ) } ∞ k =0 is a multiplier sequence although p need not have any real zeros. Also, we hasten tonote that the Hadamard product , { p ( k ) k ! } ∞ k =0 , of the two multiplier sequences { p ( k ) } ∞ k =0 and { k ! } ∞ k =0 is againa multiplier sequence. In particular, it follows that P ∞ k =0 p ( k ) k ! k ! x k ∈ L − P + . We next consider the followingquery. Given a fixed c >
0, does the entire function f ( x ) := (1 + cx ) I (2 √ x ) + √ xI (2 √ x ), where I p ( x )denotes the modified Bessel function of the first kind of order p (see equation (1.4)), have only real (positive)zeros? Calculating the Taylor coefficients of the entire function f ( x ) (of order 1 / f ( x ) = ∞ X k =0 p ( k ) k ! k ! x k ∈ L − P + since { p ( k ) k ! } ∞ k =0 is a multiplier sequence for any c >
0. If p ( x ) is a quadratic polynomial with non-negativeTaylor coefficients, then as was noted in the Introduction, { p ( k ) } ∞ k =0 need not be a multiplier sequence.However, it is a noteworthy fact that { p ( k ) k ! } ∞ k =0 is always multiplier sequence, and p ( x ) := cx + ax + b ,where a, b, c ≥ Proposition 35. If p ( x ) := cx + ax + b , where a, b, c ≥ , then { p ( k ) k ! } ∞ k =0 is a multiplier sequence.Proof. (An outline.) If p ( x ) has only real zeros, the conclusion is clear, since { p ( k ) k ! } ∞ k =0 is the Hadamardproduct of two multiplier sequences. If a = 0 (so that p ( x ) := cx + ax + b , where we may assume that b = 0), then it suffices to consider b p ( bx/a ) = 1 + x + bca x . Hence we infer, from the argument used inExample 34, that { p ( k ) k ! } ∞ k =0 is a multiplier sequence. (cid:3) The foregoing simple, but instructive, examples were introduced in order to motivate the following generalproblem.
Problem 36.
Let p ( x ) := P nk =0 a k x k , where a k ≥ k ≥ p ( x ) such that (a) { p ( k ) k ! } ∞ k =0 is a multiplier sequence and (b) { p ( k ) k ! } ∞ k =0 is a CZDS (cf. Definition 4).Part (b) of Problem 36 appears to be particularly challenging. If p ( x ) ∈ L − P + ∩ R [ x ], then it followsfrom a theorem of Laguerre (see, for example, [7, Theorem 4.1 (3)]), that the sequence { p ( k ) k ! } ∞ k =0 is a CZDS.We also call attention to one of the principal results of [8, Theorem 2.13] which completely characterizesthe class of all polynomials which interpolate CZDS. Notwithstanding, these results, at this juncture, we areobliged to expose our ignorance and formulate the following tantalizing open problem. roblem 37. Is the sequence { k + k k ! } ∞ k =0 a CZDS?We next consider more complicated sequences involving the square root function (see Section 3.1). Example 38.
Let α k := e −√ k k ! and β k := e √ k k ! for k = 0 , , . . . . Then the sequence α := { α k } ∞ k =0 is not amultiplier sequence since the Jensen polynomial g ( x ) := g ( x ; α ) := X k =0 (cid:18) k (cid:19) α k x k = 1 + 3 e x + 32 e −√ x + 16 e −√ x has two non-real zeros (see the discussion after Definition 7). In particular, ψ ( x ) := P ∞ k =0 α k k ! x k = P ∞ k =0 e −√ k x k ( k !) / ∈ L − P + . On the other hand, our numerical work shows that the Jensen polynomialsof degree n (1 ≤ n ≤ { β k } ∞ k =0 , have only real zeros. Problem 39.
Determine whether the sequence { √ kk ! } is (a) a multiplier sequence (b) a CZDS (cf. Definition4).We remark that if the sequence n √ kk ! o ∞ k =0 is a multiplier sequence (or a CZDS), then the sequence { √ k (2 k )! } ∞ k =0 is also a multiplier sequence (or a CZDS). Indeed, it follows from the Legendre DuplicationFormula [2, p. 71] that √ π k Γ( k + 1 /
2) = k !(2 k )! . Now by Laguerre’s theorem ([7, Theorem 4.1 (3)]), the se-quence { k + 1 / } ∞ k =0 is a CZDS and whence the above claim follows.In light of the discussion in Example 34 and the fact that the sequence { k + 1) } ∞ k =0 is a CZDS, we expectan affirmative answer to the following question (see also Theorem 1). Problem 40.
Is it true that for every s ∈ R + , there exists an m ∈ N , such that ∞ X k =0 k s ( k !) m x k ∈ L − P + .Before stating our next problem, we pause for a moment and briefly touch upon the characterization ofentire functions in ϕ ( x ) ∈ L − P + in terms of their Taylor coefficients. To this end, we consider the entirefunction(5.1) ϕ ( x ) := ∞ X k =0 α k x k , where α k = γ k k ! , γ = 1 , γ k ≥ k = 1 , , . . . ) . and recall the following definition. Definition 41.
A real sequence { α k } ∞ k =0 , α = 1, is said to be a totally positive sequence , if the infinitelower triangular matrix A = ( α i − j ) = α · · · α α · · · α α α · · · α α α α · · · ... ... ... ... ... . . . ( i, j = 1 , , , . . . ) , is totally positive; that is, all the minors of A of all orders are non-negative.In [1], M. Aissen, A. Edrei, I. J. Schoenberg and A. Whitney characterized the generating functions oftotally positive sequences. A special case of their result is the following theorem. Theorem 42. ( [1, p. 306] ) Let ϕ ( x ) be the entire function defined by (5 . . Then { α k } ∞ k =0 is a totallypositive sequence if and only if ϕ ( x ) ∈ L − P + . Preliminaries aside, we are now in position to state and analyze the next open problem. roblem 43. Let γ k := 1( k + 1) k +1 , k ∈ N . Determine whether the sequence (cid:8) γ k k ! (cid:9) ∞ k =0 is (a) a multipliersequence (b) a CZDS.We first claim that the entire function ψ ( x ) := P ∞ k =0 γ k x k / ∈ L − P + . In order to verify that ψ ( x ) is notin the Laguerre-P´olya class, we invoke Theorem 42 and show that { γ k } ∞ k =0 is not a totally positive sequence.Indeed, after some experimentation, we find that the determinant of the 4 × A =
127 14 is negative: det( A ) = − (38873 / if (cid:8) γ k k ! (cid:9) ∞ k =0 is a multiplier sequence, then ϕ ( x ) := P ∞ k =0 γ k k ! x k ∈ L − P + and therefore, by Theorem 42, the sequence (cid:8) γ k k ! (cid:9) ∞ k =0 = n k +1) k +1 k ! o ∞ k =0 isa totally positive sequence. In addition, we also observe that by Stirling’s formula ([2, p. 98])1( k + 1) k +1 ∼ k + 1)! √ π e − ( k +1) √ k + 1 ( k ≫ . Thus noting again the vexing presence of the square root function, it may be instructive to compare Problem43 with Problem 39. We conclude this paper with one more open problem that (i) may be useful in thestudy of CZDS and (ii) is related to our results in Section 4.
Problem 44.
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