Cai, Xing ShiLópez García, José Luis2019-06-202019-09-1020181065-2469 (Print)1476-8291 (Electronic)10.1080/10652469.2019.1627530https://academica-e.unavarra.es/handle/2454/33465This 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.12 p.application/pdfeng© 2019 Informa UK Limited, trading as Taylor & Francis GroupHurwitz–Lerch zeta functionAsymptotic expansionSpecial functionsinfo:eu-repo/semantics/articleinfo:eu-repo/semantics/openAccess