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!
In case you meant something like:
$$\sum^{\infty}_{k=1}\left[\frac{x^{k}}{k!}\cdot\left(\sum_{j=0}^{k-1}\frac{y^{j}}{j!}\right)\right]$$
then that's obviously a different case. Sorry for bugging you about the notations, but the way you wrote it above simply does not imply any dependency. Anyway, this is equal to:
$$\sum_{k=1}^{\infty}\frac{x^{k}\cdot e^{y}\cdot\Gamma(k,y)}{k!\cdot\Gamma(k)}$$
and thus your sum simplifies to
$$S=e^{-x}\sum_{k=1}^{\infty}\frac{x^{k}\cdot Q(k,y)}{k!}.$$
which leaves us the last series... good luck in finding its closed form :-). To be honest, I haven't encountered it before.