The asymptotic expansion of a generalisation of the Euler-Jacobi series

TL;DR

This paper derives the asymptotic expansion of a generalized Euler-Jacobi series for small parameters, highlighting the algebraic and exponential components, especially when parameters are even integers, and provides numerical validation.

Contribution

It extends the asymptotic analysis of the Euler-Jacobi series to general powers and weights, revealing a finite algebraic part plus exponential terms, and introduces a transformation analogous to Poisson-Jacobi.

Findings

The expansion includes a finite algebraic series and exponentially small terms.

Numerical results confirm the accuracy of the asymptotic expansion.

A new transformation for the series is established, similar to Poisson-Jacobi.

Abstract

We consider the asymptotic expansion of the sum \[S_p(a;w)=\sum_{n=1}^\infty n^{-w}\e^{-an^p}\] as $a\rightarrow 0$ in $|\arg\,a|<\pi/2$ 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.