Asymptotics of a Mathieu-Gaussian series

TL;DR

This paper derives the asymptotic expansion of a Mathieu-Gaussian series for large parameters, revealing algebraic and exponential contributions, especially for even integer gamma, with numerical validation.

Contribution

It provides a detailed asymptotic expansion for the series, including the case of even integer gamma, combining algebraic and exponential terms, which was not previously established.

Findings

Asymptotic expansion involves zeta and hypergeometric functions

Finite algebraic terms plus exponentially small contributions for even gamma

Numerical examples confirm the accuracy of the expansions

Abstract

We consider the asymptotic expansion of the functional series \[S_{\mu,\gamma}(a;\lambda)=\sum_{n=1}^\infty \frac{n^\gamma e^{-\lambda n^2/a^2}}{(n^2+a^2)^\mu}\] for real values of the parameters $\gamma$, $\lambda>0$ and $\mu\geq0$ as $|a|\to \infty$ in the sector $|\arg\,a|<\pi/4$. For general values of $\gamma$ the expansion is of algebraic type with terms involving the Riemann zeta function and a terminating confluent hypergeometric function. Of principal interest in this study is the case corresponding to even integer values of $\gamma$, where the algebraic-type expansion consists of a finite number of terms together with a contribution comprising an infinite sequence of increasingly subdominant exponentially small expansions. This situation is analogous to the well-known Poisson-Jacobi formula corresponding to the case $\mu=\gamma=0$. Numerical examples are provided to illustrate the accuracy of these expansions.