The evaluation of infinite sums of products of Bessel functions

We examine convergent representations for the sum of Bessel functions $$\sum _{n=1}^{\mathrm{\infty }}\frac{J_{\mu }(na)J_{\nu }(nb)}{n^{\alpha }}$$ 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 $a\mathrm{\ne }b$ 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.