  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.
