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

We examine the sum of a decaying exponential (depending non-linearly on the summation index) and a Bessel function in the form $$\sum _{n=1}^{\mathrm{\infty }}{e}^{-a{n}^p}\frac{J_{\nu }(a{n}^{p}x)}{(a{n}^{p}x\mathrm{/}2{)}^{\nu }}(x>0),$$ in the limit $a\to 0$, where $J_{\nu }(z)$ is the Bessel function of the first kind of real order $\nu$ and $a$ and $p$ are positive parameters. By means of a Mellin transform approach we obtain an asymptotic expansion that enables the evaluation of this sum in the limit $a\to 0$. A similar result is derived for the sum when the Bessel function is replaced by the modified Bessel function $I_{\nu }(z)$ when $x\in (0,1)$. The case of even $p$ is of interest since the expansion becomes exponentially small in character. We demonstrate that in the case $p=2$, a result analogous to the Poisson-Jacobi transformation exists for the above sum.