I am trying to complete a proof which demands showing the equality
$$\sum_{n=0}^{\infty} \left( J_{n-1}(x) - J_{n+3}(x) \right)^2 = 1, \qquad \forall \, x\geq0$$
where $J_n$ are the Bessel functions of the first kind.
I've computed the sum numerically up to a few finite $n_{\max}$ and it seems as though the sum is indeed converging to 1 for all values of $x$ within the range [0,100]. Please see the attached plot. If anyone has a way of proving this equality I'd be very interested in learning your approach!
Solutions
