I have the infinite sum
$$\sum_{m=0}^\infty \left(\frac{(m+n)!}{n!}\right)^a \frac{(-x)^m}{m!}$$
with $a\in [0,1]$ and $n\in \mathbb{N}$ and try to calculate a closed form for it. For $a=0$ this is just the Taylor expansion of $e^{-x}$ and for $a=1$ it is the Taylor expansion of $(x+1)^{-(n+1)}$.
I already tried to interpret the sum as a Taylor expansion and was looking for a generalization of $(x+1)^{-(n+1)}$, but without success.
If there is a way, to show at least that the result is positive within the radius of convergence, it would be helpful to.