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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!

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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.

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    Thanks, Johnny! I'll persevere...2012-12-21