Exponentially small expansions associated with a generalised Mathieu series

TL;DR

This paper derives exponentially small asymptotic expansions for a generalized Mathieu series with specific parameter conditions, providing both algebraic and exponential terms, and offers closed-form evaluations when parameters are integers.

Contribution

It introduces a new asymptotic expansion method for the generalized Mathieu series, including exponentially small terms, under specific parameter conditions.

Findings

Asymptotic expansion includes finite algebraic and infinite exponentially small terms.

Closed-form evaluations are possible when parameters are positive integers.

Numerical results confirm the accuracy of the asymptotic expansions.

Abstract

We consider the generalised Mathieu series \[\sum_{n=1}^\infty \frac{n^\gamma}{(n^\lambda+a^\lambda)^\mu}\qquad (\mu>0)\] when the parameters $\lambda$ ($>0$) and $\gamma$ are even integers for large complex $a$ in the sector $|\arg\,a|<\pi/\lambda$. The asymptotics in this case consist of a {\it finite} algebraic expansion together with an infinite sequence of increasingly subdominant exponentially small expansions. When $\mu$ is also a positive integer it is possible to give closed-form evaluations of this series. Numerical results are given to illustrate the accuracy of the expansion obtained.