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

Abstract

In a previous paper by Ferreira and López [Journal of Mathematical Analysis and Applications, 298(1), 2004], the authors derived an asymptotic expansion of the Lerch's transcendent $\Phi (z,s,a)$ for large $|a|$, valid for $\mathrm{Re}(a)>0$, $\mathrm{Re}(s)>0$ and $z\in \mathbb{C}\setminus [1,\infty )$. In this paper we study the special case $z\ge 1$ not covered in the previous result, deriving a complete asymptotic expansion of the Lerch's transcendent $\Phi (z,s,a)$ for $z>1$ and $\mathrm{Re}(s)>0$ as $\mathrm{Re}(a)$ goes to infinity. We also show that when $a$ is a positive integer, this expansion is convergent for $\mathrm{Re}(z)\ge 1$. As a corollary, we get a full asymptotic expansion for the sum ${\sum }_{n=1}^m{z}^n/n^s$ for fixed $z>1$ as $m\to \infty$. Some numerical results show the accuracy of the approximation.

12 page

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