I've come across this infinite series
$$\sum_{k=0}^{\infty}\frac{z^k}{k!}J_{k+m}(z)J_{k+n}(z)$$
where m and n are integers. I suspect that there is a closed form for it but I haven't found one for it. I skimmed through Watson's book and couldn't find either. Does anyone know of a closed form for it? Thanks.