Does anyone know what the following is equal to? $$\sum^{\infty}_{n=-\infty} J_n^4(x)$$
$J_n$ is the Bessel function of the first kind of order $n$. In the quantum walk problem I'm tackling, $x$ is real and $n$ is an integer such that $J_n (x)$ is always real.
Am familiar with following identities $J_{-n} (x) = (-1)^n J_n(x)$ and $ \sum_{n=-\infty}^\infty J_n^2(x) = 1$ from grad school but not sure how to make use of these (or if they're at all useful!)