Recently, I came across the following identity among first-kind Bessel functions, namely
$$2\sum_{k=1}^{\infty}(-1)^k\,k^5\,J_{2k}(x) = \frac{x^2}{4}\left(x\,J_1(x)-J_0(x)\right)$$
It's straightforward to verify that the identity holds perturbatively up to very high orders in $x$. However, I'm curious if anyone knows how to formally prove it analytically. I've searched the mathematical literature, but I couldn't find any identities involving the multiplication of Bessel functions by a monomial of power 5, as in the current case. Any hints or suggestions would be greatly appreciate.
Many thanks