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I'm searching for an expression of the following sum (all indices are integers and all variables are positive real numbers):

$$\sum_{\lambda=-\infty}^{+\infty} A^{|l-\lambda|} B^{|\lambda|} J_{\lambda+m}(a x) J_\lambda(b x)$$

Which can be rewritten as two sums over positive integers only.

I have found a sum formula here:

$$\sum_{n=-\infty}^{+\infty} t^n J_n(x) J_n(y) = J_0 \left(\sqrt{x^X+y^2-\frac{t^2+1}{t}xy}\right)$$

But that only gets me halfway there (if that is at all any closer).

For my case here, I have the opportunity to integrate out the two bessel functions, replacing them with a polynomial in (absolute value of) $\lambda$. These expressions are lengthy, though, and I'd very much like to perform the summation first.

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It seems to me that your series is related to Graf's addition theorem. There, instead of the power of $(B/A)^\lambda$, there is a trigonometric function. However, writing a complex exponential as $e^{i\lambda\theta} = (B/A)^\lambda$ could give you the answer on the complex unit circle $|B/A|=1$. Extending away from the unit circle by analyticity may give you the full answer. The main thing that's odd about your formula is that it has $(B/A)^{|\lambda|}$ instead of $(B/A)^\lambda$ in Graf's addition theorem. But perhaps that can be fixed using some reflections of the order of the Bessel functions.

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