Property of Bessel's functions

Question

While studying the properties of Bessel's function I came across a proof as :-

$ J_0^2+2(J_1^2+J_2^2+ . . . . )=1$

I have tried this with the recurrence relations, but I didn't have any idea how to proceed.

Thankyou.

Answer 1

Do you know "Jacobi-Anger" complex Fourier expansion :

$$\mathrm e^{i x \cos t}=\sum_{n=-\infty}^\infty i^n J_n(x)\mathrm e^{i n t} \ ?$$

(see https://en.wikipedia.org/wiki/Jacobi%E2%80%93Anger_expansion and its avatars given in this article).

Now use Parseval's identity (https://en.wikipedia.org/wiki/Parseval%27s_identity).

Edit : I just noticed this answer to the very same question : https://math.stackexchange.com/q/2261873 by Jack d'Aurizio which is strikingly identical to the one I have proposed !!! Oddly, the accepted answer (with a bonus...) is very intricated.