The evaluation of infinite sums of products of Bessel functions

TL;DR

This paper derives convergent representations for infinite sums involving products of Bessel functions, facilitating their computation especially as parameters approach zero, and explores cases with modified Bessel functions and logarithmic behaviors.

Contribution

It provides new convergent formulas for sums of Bessel function products, including cases with modified Bessel functions and logarithmic singularities, enhancing computational methods.

Findings

Derived convergent series representations for Bessel function sums

Analyzed logarithmic cases when parameters approach zero

Extended results to sums involving modified Bessel functions

Abstract

We examine convergent representations for the sum of Bessel functions \[\sum_{n=1}^\infty \frac{J_\mu(na) J_\nu(nb)}{n^{\alpha}}\] for $\mu$, $\nu\geq0$ and positive values of $a$ and $b$. Such representations enable easy computation of the series in the limit $a, b\to0+$. Particular attention is given to logarithmic cases that occur both when $a=b$ and $a\neq b$ for certain values of $\alpha$, $\mu$ and $\nu$. The series when the first Bessel function is replaced by the modified Bessel function $K_\mu(na)$ is also investigated, as well as the series with two modified Bessel functions.