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A Note on the Asymptotic Expansion of the Lerch's Transcendent

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A Note on the Asymptotic Expansion of the Lerch's Transcendent Integral Transforms and Special Functions ISSN: 1065-2469 (Print) 1476-8291 (Online) Journal homepage: https://tandfonline.com/loi/gitr20 A note on the asymptotic expansion of the Lerch's transcendent Xing Shi Cai & José L. López To cite this article: Xing Shi Cai & José L. López (2019) A note on the asymptotic expansion of the Lerch's transcendent, Integral Transforms and Special Functions, 30:10, 844-855, DOI: 10.1080/10652469.2019.1627530 To link to this article: https://doi.org/10.1080/10652469.2019.1627530 © 2019 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group Published online: 10 Jun 2019. Submit your article to this journal Article views: 208 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at https://tandfonline.com/action/journalInformation?journalCode=gitr20 INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 2019, VOL. 30, NO. 10, 844–855 https://doi.org/10.1080/10652469.2019.1627530 Research Article A note on the asymptotic expansion of the Lerch’s transcendent Xing Shi Caia and José L. Lópezb aDepartment of Mathematics, Uppsala University, Uppsala, Sweden; bDepartamento de Estadística, Matemáticas e Informática and INAMAT, Universidad Pública de Navarra, Pamplona, Spain ABSTRACT ARTICLE HISTORY In Ferreira and López [Asymptotic expansions of the Hurwitz–Lerch Received 21 March 2019 zeta function. J Math Anal Appl. 2004;298(1):210–224], the authors Accepted 1 June 2019 derived an asymptotic expansion of the Lerch’s transcendent KEYWORDS (z s a) |a| a > s > z ∈ C \ ∞) , , for large , valid for 0, 0 and [1, .In Hurwitz–Lerch zeta function; this paper, we study the special case z ≥ 1 not covered in Ferreira and asymptotic expansion; López [Asymptotic expansions of the Hurwitz–Lerch zeta function. J special functions Math Anal Appl. 2004;298(1):210–224], deriving a complete asymp- totic expansion of the Lerch’s transcendent (z, s, a) for z > 1 and AMS CLASSIFICATION s > 0asa goes to infinity. We also show that when a is a posi- 11M35 tive integer, this expansion is convergent for z ≥ 1. As a corollary, m zn/ns we get a full asymptotic expansion for the sum n=1 for fixed z > 1asm →∞. Some numerical results show the accuracy of the approximation. 1. Introduction The Lerch’s transcendent (Hurwitz–Lerch zeta function) [1, §25.14(i)] is defined by means of the power series ∞ zn (z, s, a) = , a = 0, −1, −2, ..., (a + n)s n=0 on the domain |z| < 1foranys ∈ C or |z|≤1fors > 1. For other values of the variables z,s,a,thefunction(z, s, a) is defined by analytic continuation. In particular2 [ ], ∞ − − 1 xs 1 e ax (z, s, a) = dx, a > 0, z ∈ C \ [1, ∞) and s > 0. (1.1) ( ) − −x s 0 1 z e This function was investigated by Erdélyi [3, § 1.11, Equation (1)]. Although using a dif- π ferent notation z = e2 ix, it was previously introduced by Lerch [4]andLipschitz[5] in connection with Dirichlet’s famous theorem on primes in arithmetic progression. If x ∈ Z, the Hurwitz–Lerch zeta function reduces to the meromorphic Hurwitz zeta func- tion ζ(s, a) [6, § 2.3, Equation (2)], with one single pole at s = 1. Moreover, ζ(s,1) is nothing but the Riemann zeta function ζ(s). CONTACT Xing Shi Cai [email protected], [email protected] © 2019 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives License (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited, and is not altered, transformed, or built upon in any way. INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 845 Properties of the Lerch’s transcendent have been studied by many authors. Among other π results, we remark the following ones. Apostol obtains functional relations for (e2 ix, s, a) π and gives an algorithm to compute (e2 ix, −n, a) for n ∈ N intermsofacertainkindof π generalized Bernoulli polynomials [7]. The function (e2 ix, s, a) is used in [8]togen- eralize a certain asymptotic formula considered by Ramanujan. Asymptotic equalities for π some weighted mean squares of (e2 ix, s, a) are given in [9]. Integral representations, as well as functional relations and expansions for (z, s, a) may be found in [6, § 2.5]. See Erdélyi et al. [10] for further properties. Here we want to remark the following two impor- tant properties of the Lerch’s transcendent valid for x, z, s ∈ C, m ∈ N [1, § 25.14.3 and § 25.14.4]: ∞ − 1 zn 1 (z, s,1) = Lis (z) := , |z| < 1, z ns n=1 where Lis(z) is the polylogarithm function [1, § 25.12], and m−1 zn (z, s, x) = zm (z, s, x + m) + , −x ∈/ N ∪{0} (x + n)s n=0 In particular, when x = 1,thesecondpropertymaybewrittenintheform m n z m+1 η(z, s, m) := = Lis(z) − z (z, s, m + 1) . (1.2) ns n=1 The finite sum η(z, s, m) for z > 1isofinterestinthestudyofrandom records in full binary trees by Janson [11]. In a full binary tree, each node has two child nodes and each level m+1 of the tree is full. Thus Tm, a full binary tree of height m,hasn = 2 − 1nodes.Inthe random records model, each node in Tm isgivenalabelchosenuniformlyatrandomfrom the set {1, ..., n} without replacement. A node u is called a record when its label is the smallest among all the nodes on the path from u to the root node. Let h(u) be the distance from u to the root. Let X(Tm) be the (random) number of records in Tm.Thenitiseasyto seethattheexpectationofX(Tm) is simply m+1 + 1 2i η (2, 1, m + 1) 2m 1 2m n n = = = + O = + O , h(u) + 1 i + 1 2 m m2 m m2 u∈Tn i=0 (1.3) where the last step follows from elementary asymptotic computations [11, Remark 1.3]. A generalization of random records, called random k-cuts, requires a similar computation which boils down to finding an asymptotic expansion of η(2, b/k, m) for some k ∈ N and 1 ≤ b ≤ k as m →∞,see[12, § 5.3.1]. Or more generally, asymptotic expansions of the function Li (z) F(z, s, a) := (z, s, a) − s , (1.4) za that generalizes the function η(z, s, m − 1) defined in (1.2), from integer to complex val- ues of the variable m: η(z, s, m − 1) =−zmF(z, s, m). Complete asymptotic expansions, 846 X. S. CAI AND J. L. LÓPEZ including error bounds, of (z, s, a) for large a have been investigated in [2]. In particular, for a > 0, s > 0andz ∈ C \ [1, ∞), we have that, for arbitrary N ∈ N [2, Theorem 1], N−1 ( ) s n −N−s (z, s, a) = cn(z) + O a , (1.5) an+s n=0 | |→∞ ( ) = ( + )...( + − ) as a .Inthisformula, s n : s s 1 s n 1 is the Pochhammer symbol, −1 c0(z) = (1 − z) and, for n = 1, 2, 3, ..., n (−1) Li−n(z) cn(z) := . (1.6) n! From the identities (1.2) and (1.5) we have that, for all N ∈ N, z ∈/ [1, ∞) and s > 0where expansion (1.5) is valid, − m N1 ( ) z s n −N η(z, s, m − 1) = Lis(z) − cn(z) + O m , (1.7) ms mn n=0 as m →∞. Unfortunately, expansion (1.5) has not been proved for z ∈ [1, ∞),andthen,in principle, the above expansion of η(z, s, m − 1) does not hold in the domain of the variable z where the approximation of η(z, s, a) has a greater interest. Had the expansion (1.5) been proved for z ≥ 1, the asymptotic computations in (1.3) would have become unnecessary and an arbitrarily precise approximation could be achieved automatically from (1.7). However, to our surprise, it seems that the expansion (1.7) is still valid when z > 1. An argument that supports this claim is the following. On the one hand, assuming for the moment that (1.7) holds for z = 2, then m + − η(2, −1, m) = n2n = (m − 1)2m 1 + 2 + O m N , N ∈ N. n=1 On the other hand, using summation by parts [13,pp.56],itiseasytoseethat m + η(2, −1, m) = n2n = (m − 1)2m 1 + 2. n=1 Thus expansion (1.7) seems to be correct for z = 2. Numerical experiments further suggest that this is also true for other values of z > 1. Then, the purpose of this paper is to show that expansion (1.7) holds for z > 1. More generally, to derive an expansion of F(z, s, a) for large a with s > 0andz ≥ 1. 2. An expansion of (z, s, a) for large a and z ≥ 1 The main result of the paper is given in Theorem 2.3. In order to formulate Theorem 2.3, we need to consider the function − − 1 − (z e x)1 a f (z, x, a) := , (2.1) 1 − z e−x and its Taylor coefficients Cn(z, a) at x = 0. We also need the two following Lemmas. INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 847 Lemma 2.1: For a, z ∈ C and n = 0, 1, 2, ..., n ( ) 1−a cn−k z k Cn(z, a) := cn(z) − z pn(z, a), pn(z, a) := (a − 1) , (2.2) k! k=0 −1 where, for z = 1, the coefficients cn(z) have been introduced in (1.6): c0(z) = (1 − z) and, for n = 1, 2, 3, ..., n (−1) Li−n(z) cn(z) := .
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