How is proof of $J_0^2 (x)$ + $2J_1^2 (x)$ + $2J_2^2 (x)$ $+ ... = 1$ done using the Bessel's generating function? Please help

Question

$J_0^2 (x)$ + $2J_1^2 (x)$ + $2J_2^2 (x)$ $+ ... = 1$. I have managed to obtain the sine and cosine expansions of Bessel functions, however, I could not arrive at the expression in question

Answer 1

It is enough to consider the Jacobi-Anger expansion(s)

$$ e^{iz \cos\theta} = J_0(z) +2\sum_{n\geq 1} i^n J_n(z)\cos(n\theta) $$ $$ \cos(z\sin\theta) = J_0(z)+2\sum_{n\geq 1}J_{2n}(z)\cos(2n\theta) $$ $$ \sin(z\sin\theta) = 2\sum_{n\geq 1}J_{2n-1}(z)\sin((2n-1)\theta)$$ and to apply Parseval's theorem. For instance $$ \int_{-\pi}^{\pi}\sin^2(z\sin\theta)\,d\theta = 4\pi\sum_{n\geq 1}J_{2n-1}^2(z) $$ $$ \int_{-\pi}^{\pi}\cos^2(z\sin\theta)\,d\theta = 2\pi J_0(z)^2+4\pi\sum_{n\geq 1}J_{2n}^2(z) $$ hence $$ 2\pi = 2\pi J_0(z)^2 + 4\pi\sum_{n\geq 1}J_n(z)^2.$$