The evaluation of infinite sums of products of Bessel functions

Abstract

We examine convergent representations for the sum of Bessel functions \[\sum_{n=1}^\infty \frac{J_μ(na) J_ν(nb)}{n^α}\] for $\mu$, $\nu \ge 0$ and positive values of $a$ and $b$. Such representations enable easy computation of the series in the limit $a,b\to 0+$. Particular attention is given to logarithmic cases that occur both when $a=b$ and $a\ne b$ 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.

arXiv comment19 pages, 0 figures

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