How to prove this Bessel equality? [closed]
Use the following: $$ e^{ix\cos t}=\sum_{n=-\infty}^\infty J_n(x)e^{int} $$ Differentiate both sides twice with reapect to $t$: $$ -(x^2\sin^2t+ix\cos t)e^{ix\cos t}=-\sum_{n=-\infty}^\infty n^2 J_n(x)e^{int} $$ taking $t=\dfrac\pi 2$ and combine $n$ with $-n$ terms and using $J_{-n}(x)=(-1)^nJ_n(x)$ we get: $$ x^2=\sum_{n=1}^\infty n^2 J_n(x)[e^{in\pi/2}+(-1)^ne^{-in\pi/2}] $$ The expression in the square bracket vanishes for odd $n$ and is $2$ otherwise, so we get: $$ x^2=2\sum_{n=1}^\infty (2n)^2 J_{2n}(x) $$