An expansion for the sum of a product of an exponential and a Bessel function. II

TL;DR

This paper derives asymptotic expansions for sums involving exponential decay and Bessel functions as the parameter approaches zero, providing new analytical tools for such sums with potential applications in mathematical physics.

Contribution

It introduces a Mellin transform-based method to evaluate the asymptotic behavior of sums involving exponential and Bessel functions, including cases with modified Bessel functions and even powers.

Findings

Asymptotic expansion for the sum as a approaches zero.

Extension to sums with modified Bessel functions for x in (0,1).

Identification of exponentially small expansions for even p, including p=2.

Abstract

We examine the sum of a decaying exponential (depending non-linearly on the summation index) and a Bessel function in the form \[\sum_{n=1}^\infty e^{-an^p}\frac{J_\nu(an^px)}{(an^px/2)^\nu}\qquad (x>0),\] in the limit $a\to0$, where $J_\nu(z)$ is the Bessel function of the first kind of real order $\nu$ and $a$ and $p$ are positive parameters. By means of a Mellin transform approach we obtain an asymptotic expansion that enables the evaluation of this sum in the limit $a\to 0$. A similar result is derived for the sum when the Bessel function is replaced by the modified Bessel function $I_\nu(z)$ when $x\in (0,1)$. The case of even $p$ is of interest since the expansion becomes exponentially small in character. We demonstrate that in the case $p=2$, a result analogous to the Poisson-Jacobi transformation exists for the above sum.