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

Abstract

We examine the sum of a decaying exponential (depending non-linearly on the summation index) and a Bessel function in the form \[\sum_{n=1}^\infty e^{-an^p}\frac{J_\nu(an^px)}{(an^px/2)^\nu}\qquad (x>0),\] in the limit $a\to0$, 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.