Could anyone give an insight on how to prove the following formula? 


$$\sum\_{n=-\infty}^{+\infty}J\_{n}(\alpha)J\_{N+n}(\alpha)=\delta\_{N0} \, ,$$


where $N$ is an integer. I checked many references but failed to figure out the calculation method. By the way, I found this relation during some numerical calculations about Bessel functions.


Thank you very much in advance.



Indefinite summation of multiplication of two Bessel functions
This is the case $y+z=0$ of the [addition formula](https://archive.org/stream/treatiseontheory00watsuoft#page/30/mode/1up) $J\_N(y+z)=\sum\_{n\in\mathbf Z}J\_n(y)J\_{N-n}(z)$, plus the fact that $J\_k(z)=\frac1{2\pi}\int\_0^{2\pi}\cos(k\theta-z\sin\theta)\,d\theta=J\_{-k}(-z)$.


