I have found a way to sum products of bessel functions in the form $$S_\ell(x,y)=\sum_{n=-\infty}^\infty (-1)^{n+\ell} I_{\ell-2n}(x)I_n(y)$$ recursively, i.e. once $S_0(x,y)$ is found, via the differential ``ladder operator'' $\hat L_+^\ell=-\partial_x+\frac{\ell}{x}-\frac{2y}{x}\partial_y$. However, I can't find a way to find $S_0$, i.e. to solve the sum $$\sum_{n=-\infty}^\infty (-1)^n I_{2n}(x)I_n(y)$$ I've looked carefully into Watson's treatise on Bessel functions, and into other textbooks (Gradshteyn&Ryzhik and Abramowitz&Stegun) with no luck. Anyone would give me some input on how to find $S_0$?

I found a ``closed'' solution (if you want to call it that way!): $S_\ell(x,y)$ is a special case of the generalized Bessel functions, for anyone interested there is a good introduction and more references in http://arxiv.org/abs/quant-ph/0608216v1