I came across the following expression involving the product of Bessel functions
$\frac{t^2}{(t^2-v^2)^2}(tJ_1(t)J_2(v)-vJ_2(t)J_1(v))^2$
I was wondering if there was a way to express it as an infinte sum involving the product of just two Bessel functions in such a way we can recast in the following form
$\sum_{n=1}^{\infty} c_n J_{2n+1}(t)J_{2n+1}(v) \ \ $ for some $c_n$ $ \in \mathbb{R}$
Maybe using the Neumann's addition theorem or maybe a generalization of it (if it exists).
Many thanks