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) := . (2.3) n!

For n = 1, 2, 3, ..., the coefficients cn(z) maybecomputedrecursivelyintheformncn(z) = − ( ) ( ) zcn−1 z .Observethatpn z, a arepolynomialsofdegreeninthevariablea. For z = 1, Definition (2.2) must be understood in the limit sense. More precisely, C0(1, a) = 1 − aand, for n = 1, 2, 3, ...,

n Bn+1 − Bn+1(a − 1) − (n + 1)(a − 1) Cn(1, a) = , (2.4) (n + 1)! where Bn are the Bernoulli numbers and Bn(a) the Bernoulli polynomials [14, §24.2].

Proof: Formulas (2.2)–(2.3) may be derived by combining the Taylor expansions at x = 0 ( − ) − − of e a 1 x and (1 − z e x) 1. Formula (2.4) follows from the generating function of the Bernoulli polynomials Bn(a) [13, Equation (7.81)],

( − ) ∞ − e a 1 x xn 1 = Bn(a) , 1 − e−x n! n=0 the definition of the Bernoulli numbers Bn = Bn(0),andtheTaylorexpansion − −x(1−a) = ( ) =− ( ) of 1 e at x 0. The derivation of the recursion ncn z zcn−1 z is straightforward [2]. 

Apart from the explicit form of the coefficients Cn(z, a) given above, we may compute them by means of the recurrence relation given in the following lemma.

1−a Lemma 2.2: For z = 1 we have that C0(z, a) = (1 − z )/(1 − z) and, for n = 1, 2, 3, ...,

n−1 − z (−1)n k (a − 1)n Cn(z, a) = C (z, a) − . 1 − z (n − k)! k n!za k=0

For z = 1, we have C0(1, a) = 1 − a, and, for n = 1, 2, 3, ...,

+ n−1 − (a − 1)n 1 (−1)n k Cn(1, a) =− − C (1, a). (n + 1)! (n + 1 − k)! k k=0 848 X. S. CAI AND J. L. LÓPEZ

Proof: Replace ∞ ∞  k  − (−x) e x = and f (z, x, a) = C (z, a)xk k! k k=0 k=0 − − − into the identity [1 − z e x]f (z, x, a) = 1 − (z e x)1 a and equate the coefficients of equal powers of x. 

Theorem 2.3: For fixed N ∈ N, a > 1 and s > 0,

N−1 Lis(z) (s)n F(z, s, a) := (z, s, a) − = Cn(z, a) + RN(z, s, a), (2.5) za an+s n=0 with Cn(z, a) given in the previous lemmas and, for z > 1,   1−N−s −a RN(z, s, a) = O (a) + az , a →∞. (2.6)

This means that expansion (2.5) has an asymptotic character for large awhenz> 1.More- over, expansion (2.5) is convergent for a = m = 2, 3, 4, ...and z ≥ 1;i.e.RN(z, s, m) → 0 as N →∞and − ∞ m1 k  1 z (s)n F(z, s, m) :=− = Cn(z, m) , z ≥ 1. (2.7) zm ks mn+s k=1 n=0

Remark 2.1: Note that, both (z, s, a) and Lis(z) are multivalued functions of z.Inour analysis and numeric experiments, we always choose the principle branch for the vari- able z. Thus (2.5) can be transformed into an asymptotic expansion of (z, s, a) without ambiguity as follows:

N−1 Lis(z) (s)n (z, s, a) = + Cn(z, a) + RN(z, s, a). za an+s n=0

Proof: Using the integral representation (2) of (z, s, a) givenin[2] and the integral representation [1, §25.12.11] of the polylogartithm,  ∞ − z xs 1 Li (z) = dx, s ( ) x − s 0 e z we find the following integral representation of the function F(z, s, a) defined in (1.4):  ∞ Li (z) 1 − − F(z, s, a) := (z, s, a) − s = xs 1 e axf (z, x, a) dx, (2.8) a ( ) z s 0 valid for a > 0, s > 0andz ∈ C,withf (z, x, a) given in (2.1). In principle, the left-hand side of (2.8) is an analytic function of z in {z : |z| < 1; | arg z| <π}.Then,theright-hand side of this equation defines the analytic continuation of F(z, s, a) in the variable z to the cut complex plane C \ (−∞,0]. INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 849

The function f (z, x, a) has the following Taylor expansion at x = 0:

n−1 k f (z, x, a) = Ck(z, a)x + rn(z, x, a), (2.9) k=0 where the coefficients Cn(z, a) are given in (2.2)–(2.4). Replacing the function f (z, x, a) in the integral (2.8) by its Taylor expansion (2.9), and interchanging sum and integral, we obtain (2.5) with  ∞ ( ) = 1 s−1 −ax ( ) Rn z, s, a : ( ) x e rn z, x, a dx. (2.10) s 0

Using the Cauchy integral formula for the remainder rn(z, x, a),wefind  − − xn f (z, w, a) 1 − (z e w)1 a r (z, x, a) = dw, f (z, w, a) := , (2.11) n π ( − ) n − −w 2 i C w x w 1 z e where we choose C tobeaclosedloopthatencirclesthepointsw = 0andw = x,itis traversed in the positive direction, and is inside the region U :={w ∈ C, w > −W, | w| < W}, for some arbitrary but fixed W ∈ (0, 2π).(U is an infinity rectangular region around the positive real line [0, ∞) of width 2W,see[2] for further details.) Figure 1 gives an example of U and C. The function f (z, w, a) is continuous in the variable w for w ∈ U.Thesingularitiesofthis function are w = log z + 2iπn, n ∈ Z \{0} and are located outside U.Defineb := a and β := a − b and write − − − −β − −β − − −β 1 − (z e w)1 a 1 − (z e w) + (z e w) − (z e w)1 b f (z, w, a) = = 1 − z e−w 1 − z e−w −w −β −w 1−b 1 − (z e ) − −β 1 − (z e ) = + (z e w) . (2.12) 1 − z e−w 1 − z e−w We have that   − − b−2 − − − − − b−1 k 1 − (z e w)1 b (z e w) k − (z e w) k 1 ew = =− . (2.13) 1 − z e−w 1 − z e−w z k=0 k=1

Figure 1. The integration loop C and the region U. 850 X. S. CAI AND J. L. LÓPEZ

Thefollowingboundsarevalidforw ∈ U and a certain constant M0 > 0 independent of b and w:

− ( −w)−β 1 z e −w −β βw , (z e ) ≤ M0 e ,for0≤β<1, 1 − z e−w   b−1 k ew e w ≤ (b − 1) ,forw ≤ log z, z z (2.14) k=1     b−1 k b−1 ew e w ≤ (b − 1) ,forw ≥ log z. z z k=1

Therefore, from (2.12), (2.13) and (2.14) we have that, for any w ∈ U and a > 1,

− ( − ) |f (z, w, a)|≤M|a|[e w + z1 a e a 1 w], for a certain constant M > 0 independent of a and w.ThepathC in (2.11) may be chosen in such a way that w ≤ x + 1/a. Then, from (2.11), we find the following bound for the remainder rn(z, x, a):

n x 1−a x(a−1) |rn(z, x, a)|≤Mn|a|x [e + z e ], for x ≥ 0, where Mn is a certain positive constant that depends on the geometry of the path C chosen in (2.11) and a but not on a. Then, from (2.10),  | | ∞ | ( )|≤Mn a n+s−1 (1−a)x + 1−a −x = O(( )1−n−s + 1−a) Rn z, s, a |( )| x [e z e ]dx a az . s 0 This proves (2.6) and the asymptotic character of the expansion (2.5) for large a when z > 1. Finally, we prove formula (2.7). When a = m ∈{2, 3, 4, ...}, the Taylor coefficients Cn(z, m) of f (z, x, m) at x = 0 are given in (2.2)–(2.4) with a = m.Butwemayderivea simpler formula for Cn(z, m).Wehavethat

− − m−2 − − − − − m−1 1 − (z e x)1 m (z e x) k − (z e x) k 1 ekx f (z, x, m) := = =− . 1 − z e−x 1 − z e−x zk k=0 k=1

Then, the Taylor coefficients Cn(z, m) of f (z, x, m) at x = 0 are the sum of the Taylor − coefficients of z k ekx,thatis,

− m1 n   1 k 1 −m −1 −1 Cn(z, m) =− = z (z , −n, m) − Li−n(z ) . n! zk n! k=1 Since m−1 m−1 + 1 kn (m − 1)n 1 (m − 1)n 1 |Cn(z, m)|= ≤ ≤ , n! zk n! zk n! k=1 k=1 INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 851 we have that    ∞ ∞ − ∞ ( +) − n n+s−1 −mx m 1 n s m 1 |Cn(z, m)| x e dx ≤ < ∞. ms n! m n=0 0 n=0

Therefore, when a = m ≥ 2, we may replace f (z, x, m) into the integral (2.8) by its Taylor expansion (2.9) and interchange sum and integral. This proves (2.7). 

Corollary 2.4: For fixed N ∈ N, z > 1, m ∈ N and s > 0,

η(z, s, m − 1)

− m N1   ( ) z 1−m s n 1+s−N s+1 −m =− cn(z) + z pn(z, m) + O m + m z , ms mn n=0 as m →∞.Moreover, for z ≥ 1,

∞ m    ( ) z 1−m s n η(z, s, m − 1) =− cn(z) + z pn(z, m) . ms mn n=0

3. Final remarks and numeric experiments Remark 3.1: The integral representation (1.1) of (z, s, a) is not valid for z ∈ [1, ∞) because of the pole of the integrand at x = log z. This pole is removed by the subtraction − − of the function xs 1(ex − z) 1 to the integrand. We obtain in this way the integral repre- −a sentation (2.8) of the function F(z, s, a) := (z, s, a) − z Lis(z),freeofthepolex = log z and valid for z ∈ C \ (−∞,0].

Remark 3.2: Since f (z, x,0) ≡ 1, we have C0(z,0) = 1andCn(z,0) ≡ 0forn = 1, 2, 3, ... Thus by (2.2), for all n ∈ N and z = 1,

− n 1 n−k z (−1) ck(z) cn(z) = , 1 − z (n − k)! k=0 which is equivalent to   n−1 z z n Li−n(z) = + Li− (z). (1 − z)2 1 − z k k k=1

Remark 3.3: Observe that the terms of the expansion (2.5) are not a pure Poincaré expan- −k−s 1−a sion in the asymptotic sequence a , as the coefficients Ck(z, a) = ck(z) + z pk(z, a), z ≥ 1, depend on a (pk(z, a) isapolynomialofdegreek in a). For z > 1, these coefficients 852 X. S. CAI AND J. L. LÓPEZ are separable and then we may write the expansion (2.5) in the form n−1 ( ) n−1 ( ) s k 1−a s k F(z, s, a) = ck(z) + z pk(z, a) + Rn(z, s, a) ak+s ak+s k=0 k=0 n−1 ( ) s k 1−n−s 1−a = ck(z) + O((a) + az ). ak+s k=0 The first expansion is the expansion (2.5) of (z, a, s),validforz ∈/ [1, ∞).Thesecond one is an exponentially small correction; when z is very large, it is a small correction, but when z is close to 1, it is not negligible.

Remark 3.4: Using the summation by parts formula [15, §2.10.9], we have that

m+1 m   k z 1 z z − −s z η(z, s, m) = − + 1 − 1 + k 1 − ( + )s − − s z 1 m 1 z 1 z 1 = k k 1   ∞ m+1 m  ( ) (− )n−1 k = z 1 − z + z s n 1 1 z z − 1 (m + 1)s z − 1 z − 1 n! kn ks k=1 n=1 + ∞ − zm 1 1 z z (s) (−1)n 1 = − + n η(z, s + n, m). z − 1 (m + 1)s z − 1 z − 1 n! n=1 Thus expansion (1.7) could also be proved by induction on N using the above identity.

Remark 3.5: Note that, for z = 1, m−1 ∞ ∞ 1 1 1 −F(1, s, m) = η (1, s, m − 1) = = − = ζ(s) − ζ(s, m), ns ns ns n=1 n=1 n=m where ζ(s) denotes the Riemannzetafunction[1, §25.2] and ζ(s, m) denotes the Hurwitz zeta function [1, §25.11]. Thus (2.7) gives a new series representation of ζ(s, m) for m ∈ N, as ∞ (s) ζ(s, m) = ζ(s) + C (1, m) k k k+s = m k 0   ∞ 1 B + − B + (m − 1) − (k + 1)(m − 1)k (s) = ζ(s) + 1 − m + k 1 k 1 k s ( + ) k m = k 1 ! m  k 1  ∞ ∞ 1 B (s) − B + (m − 1) (s) = ζ(s) + 2 − m − ms + 2k 2k 1 − k 1 k . ms (2k)! m2k−1 (k + 1)! mk k=1 k=1 On the other hand, using the Euler–Maclaurin’s summation formula [13, §9.5], or the asymptotic expansion of ζ(s, m) in [1, Equation (25.11.43)], we have, for s > 1, 1−s −s n ( ) m m B2k s 2k−1 ˆ F(1, s, m) =−ζ(s) + + + + Rn(s, m), s − 1 2 (2k)! m2k+s−1 k=1 INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 853

Table 1. The relative error in the approximation (2.5) for Lerch’s transcendent. z = 2, s = 1 na= 5 a = 10 a = 20 5 7.87e−2 2.22e−2 6.21e−4 10 2.13e−2 7.55e−3 7.36e−5 15 6.69e−3 3.68e−3 3.24e−5 z = 5, s = 2 na= 5 a = 10 a = 20 5 8.36e−2 2.57e−3 5.87e−5 10 2.82e−2 2.89e−4 1.21e−7 15 1.13e−2 1.23e−4 2.66e−9 z = 2, s = 2 na= 10+ia= 30+ia= 50+i 5 1.60e−1 3.67e−4 2.41e−5 10 9.14e−2 1.11e−5 3.32e−8 15 5.92e−2 3.62e−6 4.75e−10 z = 5, s = 3 na= 10+ia= 30+ia= 50+i 5 9.59e−3 2.52e−5 1.91e−6 10 2.37e−3 1.04e−8 5.78e−11 15 1.43e−3 3.47e−11 1.35e−14

Figure 2. The blue line is the graphic of the function F(z, s, a), whereas the red, gold and green functions represent the right-hand side of (2.5) for increasing values of the approximation order n. 854 X. S. CAI AND J. L. LÓPEZ with | | | ( ) | ˆ B2n+2 s 2n+1 Rn(s, m) ≤ . (2n + 2)! m2n+s+1

The following tables and pictures show some numerical experiments about the accuracy of the approximations given in Theorem 2.3. In the tables we compute the absolute value ¯ of the relative error Rn(z, s, a) in the approximation (2.5), defined in the form  n−1 (s)k C (z, a) + ¯ k=0 k ak s Rn(z, s, a) := 1 − . F(z, s, a) In Table 1 and Figure 2, we evaluate the Lerch’s transcendent, the Polylogarithm and all the approximations with the symbolic manipulator Wolfram Mathematica 10.4.

Acknowledgments The referee is acknowledged for his/her useful comments and improving suggestions. In particular, for suggesting the following alternative approach to derive our main result. Integral (1.1) can be writtenintheform     ∞ s−1 −ax ∞ 1 x e 1 − − 1 1 (z, s, a) = dx + xs 1 e ax − dx. ( ) − ( ) − −x − s 0 x log z s 0 1 z e x log z An asymptotic expansion of the second integral for large a may be derived by using Watson’s lemma. The first integral can be written as an incomplete gamma function and takes over the role ofthe polylogarithm in our approach.

Disclosure statement No potential conflict of interest was reported by the authors.

Funding This work was supported by the Knut and Alice Wallenberg Foundation under a Grant for a post- doctoral position in Sweden for foreign researchers and Ministerio de Economía y Competitividad of the Spanish government under Grant MTM2017-83490-P.

References [1] Apostol TM. Zeta and related functions. In: NIST handbook of mathematical functions. Cambridge: Cambridge University Press; 2010. Chapter 25. p. 601–616. [2] Ferreira C, López JL. Asymptotic expansions of the Hurwitz–Lerch zeta function. J Math Anal Appl. 2004;298(1):210–224. [3] Bateman H. Higher transcendental functions. New York: McGraw-Hill; 1953.  π K( ) = ∞ e2k ix [4] Lerch M. Note sur la fonction w, x, s k=0 (w+k)s .ActaMath.1887;11(1–4):19–24. [5] Lipschitz R. Untersuchung einer aus vier Elementen gebildeten Reihe. J Reine Angew Math. 1857;54:313–328. [6] Srivastava HM, Choi J. Series associated with the zeta and related functions. Dordrecht: Kluwer Academic Publishers; 2001. [7] Apostol TM. On the Lerch zeta function. Pacific J Math. 1951;1:161–167. [8] Katsurada M. On an asymptotic formula of Ramanujan for a certain theta-type series. Acta Arith. 2001;97(2):157–172. INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 855

[9] Klusch D. Asymptotic equalities for the Lipschitz–Lerch zeta-function. Arch Math. 1987;49(1):38–43. [10] Erdélyi A, Magnus W, Oberhettinger F, et al. Higher transcendental functions. Vols. I, II. New York–Toronto–London: McGraw-Hill Book Company, Inc.; 1953. Based, in part, on notes left by Harry Bateman. [11] Janson S. Random records and cuttings in complete binary trees. In: Mathematics and com- puter science. III. Basel: Birkhäuser; 2004. p. 241–253. (Trends math.). [12] Cai XS, Devroye L, Holmgren C, et al. Cutting resilient networks, 2018 Apr. ArXiv e-prints. [13] Graham RL, Knuth DE, Patashnik O. Concrete mathematics. 2nd ed. Reading (MA): Addison- Wesley Publishing Company; 1994. A foundation for computer science. [14] Dilcher K. Bernoulli and Euler polynomials. In: NIST handbook of mathematical functions. Cambridge: Cambridge University Press; 2010. Chapter 24. p. 587–599. [15]OlverFWJ,WongR.Asymptoticapproximations.In:NISThandbookofmathematicalfunc- tions. Cambridge: Cambridge University Press; 2010. Chapter 2. p. 41–70.