Closed form expression for $\sum_{n=0}^{\infty} J_n^2(x) \cos(ny)$, where $J_n(x)$ is the Bessel function of order $n$

Question

Anyone can find/calculate a closed form expression for the sum $$ \sum_{n=0}^{\infty} J_n^2(x) \cos(ny), $$ where $J_n$ is the Bessel function?

Answer 1

Just solve for the sum in the addition formula (31) of Neumann 1867, p. 65 (also in Watson p. 128, or more conceptually Vilenkin 1968, formula (4) p. 209): $$ J_0\left(2r\sqrt{\frac{1-\cos\theta}2}\right) = J_0(r)^2 + 2\sum_{n=1}^\infty J_n(r)^2\cos(n\theta). $$