I am interested in evaluating the following infinite sum \begin{equation} \sum_{n=0}^{\infty} \frac{\alpha^{n}}{n!}n^{\beta} \end{equation}
where both $\alpha$ and $\beta$ are real numbers. However, in addition, $\alpha$ is always positive.
Clearly the sum converges for any value of $\alpha$ and $\beta$ since the factorial kills both exponential and power terms for sufficiently large $m$'s. Does the sum have a closed form?
Assuming $\alpha=e^{\gamma}$,
$$\sum_{n\geq 0}\frac{\alpha^n}{n!}n^\beta = \sum_{n\geq 0}\frac{e^{\gamma n}}{n!}n^{\beta}=\left.\frac{\partial^\beta}{\partial u^{\beta}}\,\exp(\exp( u))\right|_{u=\gamma}.$$ If $\beta\not\in\mathbb{N}$, see fractional calculus.