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Infinite sum involving powers and factorials
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, ...
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If the Infinite sum of a series is known, what is the sum of element wise product with another series?
Suppose we know the summation of some series $G(n)$ such that, $$\sum_{n=1}^{\infty}G(n)=S_1.$$
Now assume another summation $S_2$ is expressed as,
$$S_2=\sum_{n=1}^\infty G(n) e^{i\frac{2\pi}{m}n}; \...
