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

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 $\mathrm{\Phi }(z,s,a)$ for large $\mathrm{\mid }a\mathrm{\mid }$, valid for $\mathrm{R}\mathrm{e}(a)>0$, $\mathrm{R}\mathrm{e}(s)>0$ and $z\in \mathbb{C}\setminus [1,\mathrm{\infty })$. 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 $\mathrm{\Phi }(z,s,a)$ for $z>1$ and $\mathrm{R}\mathrm{e}(s)>0$ as $\mathrm{R}\mathrm{e}(a)$ goes to infinity. We also show that when $a$ is a positive integer, this expansion is convergent for $\mathrm{R}\mathrm{e}(z)\ge 1$. As a corollary, we get a full asymptotic expansion for the sum ${\sum }_{n=1}^m{z}^n\mathrm{/}{n}^s$ for fixed $z>1$ as $m \to \infty$. Some numerical results show the accuracy of the approximation.