I'm wondering if the following series has a closed form:
$S = e^{-x} \sum_{k=1}^{\infty} \left[\frac{x^k}{k!} \cdot \left(e^{-y} \sum_{l=0}^{k-1} \frac{y^l}{l!} \right) \right]$
I occasionally stumble across it when I play around with integrals involving modified Bessel functions of the first kind, and it's becoming a bit of a nuisance.
I've checked Prudinkov's Integrals and Series, Vol. 2 and the NIST handbook, but I've had no luck.. If it helps, the inner series is a regularized incomplete gamma function, i.e. $\frac{\Gamma(k, y)}{\Gamma(k)} = e^{-y} \sum_{l=0}^{k-1} \frac{y^l}{l!}$.
All advice much appreciated!