A note on the asymptotic expansion of the Lerch’s transcendent

This is an accepted manuscript of an article published by Taylor & Francis in Integral Transforms and Special Functions on 10 Jun 2019, available online: https://doi.org/10.1080/10652469.2019.1627530In Ferreira and López [Asymptotic expansions of the Hurwitz–Lerch zeta function. J Math Anal Appl. 2004;298(1):210–224], the authors derived an asymptotic expansion of the Lerch's transcendent Φ(z,s,a) for large |a|, valid for Ra>0, Rs>0 and z∈C∖[1,∞). In this paper, we study the special case z≥1 not covered in Ferreira and López [Asymptotic expansions of the Hurwitz–Lerch zeta function. J Math Anal Appl. 2004;298(1):210–224], deriving a complete asymptotic expansion of the Lerch's transcendent Φ(z,s,a) for z>1 and Rs>0 as Ra goes to infinity. We also show that when a is a positive integer, this expansion is convergent for Rz≥1. As a corollary, we get a full asymptotic expansion for the sum ∑mn=1zn/ns for fixed z>1 as m→∞. Some numerical results show the accuracy of the approximation.This work is supported by the Knut and Alice Wallenberg Foundation and the Ministerio de Economía y Competitividad of the spanish government (MTM2017-83490-P)