If $J\_n(x)$ is the Bessel function of order $n$, we know that for all $x$, $$\sum\_{n=-\infty}^{\infty} J\_n^2(x)=J\_0^2(x)+2\sum\_{n=1}^{\infty} J\_n^2(x)=1.$$
What is known about
$$
\sum\_{n=-\infty}^{\infty} J\_n^4(x) ?
$$
It is smaller than $1$, but is there a nice lower bound?



Estimate on sum of $J_n^4$
The paper [here](http://inside.mines.edu/~pamartin/ref-paps/R101_JPAw.pdf) mentions the integral $\displaystyle\sum\_{n = -\infty}^\infty J\_n^4(x) = \frac{2}{\pi} \int\_0^\frac{\pi}{2} J\_0^2(2x \sin \theta) d\theta$ 
and the asymptotic $\displaystyle \sum\_{n = -\infty}^\infty J\_n^4(x) = \frac{1}{x\pi^2}(\log x + 5 \log 2 + \gamma) + O(x^{-\frac{3}{2}})$ as $x \to \infty$, and gives references.


