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I am trying to compute the following double sum:

[itex]\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}[/itex]

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

[itex]\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][/itex]

where [itex]_2F_1[/itex] 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!

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# Stuck - infinite sum

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