I'm working on a problem that involves the integrals of various Bessel functions that Mathematica can't symbolically handle. I've managed to grind out the transformations and integrals by hand, and everything seems to simplify nicely except for a single sum involving the upper incomplete Gamma functon that I cannot get into closed form, even though it seems like there should be one. It's
$$S(x)=\sum_{n=1}^\infty \frac{\Gamma(n,x)}{[\Gamma(n)]^2}$$
I've checked several resources and I can't seem to find any mention of this. I was wondering if anybody might be familiar with a closed-form solution or a special function with an equivalent definition.
Thanks!