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

In Ferreira and Lopez [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 Phi(z, s, a) for large vertical bar a vertical bar, valid for Ra > 0, Rs > 0 and z is an element of C \ [1, infinity). In this paper, we study the special case z >= 1 not covered in Ferreira and Lopez [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 Phi(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 Sigma(m)(n=1) z(n)/n(s) for fixed z > 1 as m -> infinity. Some numerical results show the accuracy of the approximation