The evaluation of infinite sums of products of 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.