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

## TL;DR

This paper derives convergent asymptotic expansions for sums involving exponential and Bessel functions, useful for evaluating these sums as parameters approach specific limits, with extensions to modified Bessel functions and alternating sums.

## Contribution

It introduces a novel double Mellin-Barnes integral method to obtain convergent asymptotic expansions for sums of exponential and Bessel functions, including modified and alternating cases.

## Key findings

- Derived convergent asymptotic expansion for the sum with Bessel functions.
- Extended the method to sums with modified Bessel functions.
- Provided analysis for the sum's behavior as parameter a approaches zero.

## Abstract

We examine convergent representations for the sum of a decaying exponential and a Bessel function in the form \[\sum_{n=1}^\infty \frac{e^{-an}}{(\frac{1}{2} bn)^\nu}\,J_\nu(bn),\] where $J_\nu(x)$ is the Bessel function of the first kind of order $\nu>-1/2$ and $a$, $b$ are positive parameters. By means of a double Mellin-Barnes integral representation we obtain a convergent asymptotic expansion that enables the evaluation of this sum in the limit $a\to 0$ with $b<2\pi$ fixed. A similar result is derived for the sum when the Bessel function is replaced by the modified Bessel function $K_\nu(x)$. The alternating versions of these sums are also mentioned.