A Poisson-Jacobi-type transformation for the sum $\sum_{n=1}^\infty n^{-2m} \exp (-an^2}$ for positive integer $m$

TL;DR

This paper derives an asymptotic expansion for a class of sums involving exponential decay and power terms, generalizing the Poisson-Jacobi transformation for specific integer powers, with numerical validation.

Contribution

It introduces a new asymptotic expansion for sums of the form  n^{-w} e^{-an^2} as a approaches zero, extending the Poisson-Jacobi transformation to cases with w=2m.

Findings

Derived an asymptotic expansion for the sum as a 0

Extended Poisson-Jacobi transformation to w=2m cases

Numerical results confirm the expansion's accuracy

Abstract

We obtain an asymptotic expansion for the sum \[S(a;w)=\sum_{n=1}^\infty \frac{e^{-an^2}}{n^{w}}\] as $a\rightarrow 0$ in $|\arg\,a|<\pi/2$ for arbitrary finite $w>0$. The result when $w=2m$, where $m$ is a positive integer, is the analogue of the well-known Poisson-Jacobi transformation for the sum with $m=0$. Numerical results are given to illustrate the accuracy of the expansion.