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

In 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 for large , valid for , and. In this paper, we study the special case 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 for z>1 and as goes to infinity. We also show that when a is a positive integer, this expansion is convergent for. As a corollary, we get a full asymptotic expansion for the sum for fixed as. Some numerical results show the accuracy of the approximation. [ABSTRACT FROM AUTHOR]