Asymptotic expansion of Mathieu-Bessel series. II

TL;DR

This paper derives the asymptotic expansion of the Mathieu-Bessel series for large parameters using Mellin transforms, providing integral representations, residue calculations, and numerical validation.

Contribution

It introduces a Mellin transform-based method to obtain asymptotic expansions of Mathieu-Bessel series, including an alternating variant, with integral formulas and numerical illustrations.

Findings

Derived asymptotic expansions for large |a|

Provided integral representations involving the Riemann zeta function

Validated expansions with numerical examples

Abstract

We consider the asymptotic expansion of the Mathieu-Bessel series \[S_{\nu,\gamma}^{\mu}(a,b)=\sum_{n=1}^\infty \frac{n^\gamma K_\nu(nb/a)}{(n^2+a^2)^\mu}, \qquad (\mu>0, \nu\geq 0, b>0, \gamma\in {\bf R})\] as $|a|\to\infty$ in $|\arg\,a|<\pi/2$ with the other parameters held fixed, where $K_\nu(x)$ is the modified Bessel function of the second kind of order $\nu$. We employ a Mellin transform approach to determine an integral representation for $S_{\nu,\gamma}^{\mu}(a,b)$ involving the Riemann zeta function. Asymptotic evaluation of this integral involves appropriate residue calculations. Numerical examples are presented to illustrate the accuracy of each type of expansion obtained. The expansion of the alternating variant of $S_{\nu,\gamma}^\mu(a,b)$ is also considered.