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

TL;DR

This paper extends the asymptotic expansion of Lerch's transcendent to the case where z ≥ 1, providing a complete expansion for z > 1 and Re(s) > 0, with applications to sums involving z^n/n^s.

Contribution

It derives a full asymptotic expansion of Lerch's transcendent for z ≥ 1, including the case z > 1, and shows convergence when a is a positive integer.

Findings

Derived a complete asymptotic expansion for z > 1 and Re(s) > 0.

Proved convergence of the expansion when a is a positive integer.

Provided numerical evidence of the approximation's accuracy.

Abstract

In a previous paper by Ferreira and L\'opez [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 $\vert a\vert$, 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.