Recently, I came across the following identities among first-kind Bessel functions, namely

$2\sum_{k=1}^{\infty}(-1)^k\,k^5\,J_{2k}(x) = \frac{x^2}{4}\left[x\,J_1(x)-J_0(x)\right] \,\, \,(1)$

and

$2\sum_{k=1}^{\infty}\,(-1)^k\,k^3\,J_{2k}(x) = -\frac{x^2}{4}J_0(x) \, \, \, (2)$

It's straightforward to verify that the identities hold perturbatively up to very high orders in $x$. However, I'm curious if anyone knows how to formally prove them analytically. I've searched the mathematical literature, but I couldn't find any identities involving the multiplication of even Bessel functions by a monomial of power 5 or 3, as in (1) and (2). Any hints or suggestions would be greatly appreciate.

Many thanks in advance

Recently, I came across the following identities among first-kind Bessel functions, namely $$ 2\sum_{k=1}^{\infty}(-1)^k\,k^5\,J_{2k}(x) = \frac{x^2}{4}\left[x\,J_1(x)-J_0(x)\right] \label{1}\tag{1} $$ and $$ 2\sum_{k=1}^{\infty}\,(-1)^k\,k^3\,J_{2k}(x) = -\frac{x^2}{4}J_0(x) \label{2} \tag{2}. $$ It's straightforward to verify that the identities hold perturbatively up to very high orders in $x$. However, I'm curious if anyone knows how to formally prove them analytically. I've searched the mathematical literature, but I couldn't find any identities involving the multiplication of even Bessel functions by a monomial of power 5 or 3, as in \eqref{1} and \eqref{2}. Any hints or suggestions would be greatly appreciate.

Many thanks in advance