The Asymptotic Expansion of a Generalisation of the Euler-Jacobi Series

Abstract

We consider the asymptotic expansion of the sum
$$S_p(a;w)=\sum _{n=1}^{\mathrm{\infty }}\frac{e^{-a{n}^p}}{n^w}$$
as $a\to 0$ in $|\arg a|<\text{fs}\pi$ for arbitrary finite $p>$ and $w>0$.
Our attention is concentrated mainly on the case when $p$ and $w$ are both even integers, where the expansion consists of a {\it finite} algebraic expansion together with a sequence of increasingly subdominant exponential expansions. This exponentially small component produces a transformation for $S_p(a;w)$ analogous to the well-known Poisson-Jacobi
transformation for the sum with $p=2$ and $w=0$.  Numerical results are given to illustrate the accuracy of the expansion obtained.