Riemann Zeta Function, Stirling's Numbers, and Infinite Series A while back I was able to prove the following identity,
$$\sum_{k=1}^{\infty}\frac{\Gamma(k+r)}{\Gamma(k)(k+r)^s}=\sum_{k=0}^{r}s(r+1,r+1-k)\zeta(s-r+k)$$
where $s(k,n)$ are the Sterling numbers of the first kind. To my satisfaction (and dismay) I was able to type this infinite series into WolframAlpha and it gave me my exact same finite summation to the point, meaning one I was correct but two it had already been done. After some googling I was seeing quite a few papers that tied finite sums involving Sterlings numbers and the Riemann Zeta function to infinite sums involving the Pochhammer symbol, $(x)_k$,
$$(x)_k=\frac{\Gamma(x+k)}{\Gamma(x)}$$
I was wondering if anyone knew whether these results are well known and/or what central theorem/idea they stem from.
 A: Suppose we seek to simplify the sum
$$\sum_{k=1}^\infty \frac{\Gamma(k+r)}{\Gamma(k) (k+r)^s}.$$
with $r$ a positive integer. This is
$$\sum_{k=1}^\infty \frac{(k+r-1)!}{(k-1)! (k+r)^s}
= \sum_{k=1}^\infty \frac{1}{(k+r)^s} (k+r-1)^{\underline{r}}
\\ = \sum_{k=1}^\infty 
\frac{1}{(k+r)^{s+1}} (k+r)^{\underline{r+1}}.$$
Now recall that
$$x^{\underline{r}} = \sum_{q=0}^r 
(-1)^{r-q} \left[r\atop q\right] x^q.$$
We thus obtain
$$\sum_{k=1}^\infty 
\frac{1}{(k+r)^{s+1}} \sum_{q=0}^{r+1}
(-1)^{r+1-q} \left[r+1\atop q\right] (k+r)^q
\\ = \sum_{q=0}^{r+1}
(-1)^{r+1-q} \left[r+1\atop q\right] 
\left(\zeta(s+1-q) - \sum_{p=1}^r \frac{1}{p^{s+1-q}}\right).$$
Observe however that
$$ \sum_{q=0}^{r+1}
(-1)^{r+1-q} \left[r+1\atop q\right] 
\sum_{p=1}^r \frac{1}{p^{s+1-q}}
\\ = \sum_{p=1}^r \frac{1}{p^{s+1}}
\sum_{q=0}^{r+1}
(-1)^{r+1-q} \left[r+1\atop q\right] p^q
\\ = \sum_{p=1}^r \frac{1}{p^{s+1}} p^{\underline{r+1}}.$$
However with  $1\le p\le r$ the term  $p^{\underline{r+1}}$ includes a
zero factor and hence the latter sum vanishes, leaving just
$$\sum_{q=0}^{r+1}
(-1)^{r+1-q} \left[r+1\atop q\right] \zeta(s+1-q).$$
To ensure convergence we must have $s-r\gt 1.$
Wikipedia lists the
Stirling number identity.
