# Expression for infinite sum of two Bessel functions and a power

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.

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.