A quick question, anyone recognizes how they are related? \begin{equation} \lim_{\epsilon \to 0+} \left[\int_{\epsilon}^\infty (1-e^{-t})^{-\alpha-1} (-t)^\beta e^{-t} dt\right] = \sum_{j=1}^\infty \frac{(\alpha+1)_j}{j!} \lim_{\epsilon \to 0+} \int_\epsilon^\infty e^{-(j+1)t} t^\beta dt \end{equation} where the $(a)_k = a(a+1)\cdots(a+k-1)$ is the Pochhammer symbol.
My best guess is taylor series, but I couldn't derive the general formula somehow....
As noted, this is just the generalized binomial theorem which can be interpreted as the Maclaurin expansion of the following:
$$(1+t)^\alpha=\sum_{k=0}^\infty\binom\alpha kt^k$$
where $\displaystyle\binom\alpha k=\frac{(\alpha-k+1)_k}{k!}$, replacing $\alpha\to-\alpha-1$ and $t\to-e^{-t}$ to get
$$(1-e^{-t})^{-\alpha-1}=\sum_{k=0}^\infty\binom{-\alpha-1}k(-1)^ke^{-kt}=\sum_{k=0}^\infty\frac{(\alpha+1)_k}{k!}e^{-kt}$$
It would also appear that there is a missing $(-1)^\beta$ and the sum index should start at $j=0$.