I want to evaluate the sum: $$\sum_n e^{inz}J_n(x)J_{m-2n}(y)$$ And I wonder if there is any known identity that can help me make itt analytical. I'm aware of some other similar Bessel identities, like Graf's and Gegenbauer's summation rules $$\sum_n e^{inz}J_n(x)J_{n+m}(y)=J_m(w)e^{imP}$$ Where P and m have some geometrical interpretation, and also $$\sum_n J_n(x)J_{m\pm n}(y)=J_m(x\pm y)$$ Guess they might be useful but I still don't see a way to evaluate the first sum. If someone have an Idea I'll be grateful :)
2026-04-01 05:37:26.1775021846
Summing bessel functions
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