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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

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What exactly do you want to know about $S_0$? What does it mean to "solve" the sum? – Matt Young Jul 6 '11 at 16:29
Mathematica does not give a closed form expression for $S_0$. And if you can't find one in Watson, is there reason to think there should be one? – Stopple Jul 6 '11 at 17:16
@Matt: what I meant was to find a closed form expression for $S_0$. @Stopple: no, you're right, there's no reason, but you never know..! Watson's book was written in the 60s, there might be something new hiding around. Especially because it looks so simple! – Ziofil Jul 27 '11 at 10:21

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