Remarks on some integrals and series involving the Stirling numbers and $\zeta(n)$

From the perspective of the well-known identity

$${}_2{F_1}(a,b;c;1) = \frac{{\Gamma (c)\Gamma (c - a - b)}} {{\Gamma (c - a)\Gamma (c - b)}},$$

we clarify the connections between the Stirling numbers $s_k^n$ and the Riemann zeta function $\zeta (n)$. As a consequence, certain series and integrals can be evaluated in terms of $\zeta (n)$ and $s_k^n$.