The following integral equation was presented back in the late 30's by Watson and Szego (Journal of the London Mathematical Society) but I cannot access the journal. Any ideas on a proof ?
$$e^{-x} x^{\alpha } L_n^{\alpha }(x){}^2=\int_0^{\infty } J_{2 \alpha }\left(2 \sqrt{x y}\right) e^{-y} y^{\alpha } L_n^{\alpha }(y){}^2 \, dy$$
where $L_n^{\alpha }(x)$ is the generalized Laguerre polynomial and $J_{2 \alpha }(z)$ is the Bessel function of the first kind.