# Stuck - infinite sum

1. Dec 7, 2011

### Orbb

Hello Physicsforum,

I am trying to compute the following double sum:

$\sum_{j\in\mathbb{N}_0/2}\sum_{m=-j}^j\frac{x^{j+m}}{(j+m)!(j-m)!}e^{-\kappa^2j(j+1)/s}$

where x, kappa and s are parameters. It is possible with e.g. Mathemtatica to carry out the sum over m explicitly, which yields

$\sum_{j\in\mathbb{N}_0/2}(j!)^{-2}e^{-\kappa^2j(j+1)/s}[_2F_1(1,-j,j+1,-x^{-1})+_2F_1(1,-j,j+1,-x)-1]$

where $_2F_1$ is the ordinary hypergeometric function. This is however a fairly horrendous expression to sum over. It would be intereseting enough to understand the asymptotic behaviour of the final result for large and for small x as a function of s and kappa.

Does anybody have ideas/tricks in mind how to deal with this sum and maybe approximate it?

Any suggestions would be much appreciated!

2. Feb 4, 2012

### Orbb

Anyone an idea? Sorry for bumping this.

3. Feb 5, 2012

### HallsofIvy

Part of the problem is that this has nothing to do with "Linear and Abstract Algebra". I am moving it to "general math".