Factorial Moments of the Geometric Distribution of Order $k$

We derive a simple expression for the $r^{th}$ factorial moment ${\mu }_{(r)}$ of the geometric distribution of order $k$ with success parameter $p\in (0,1)$ (and $q=1-p$) in terms of its probability mass function $f_k(n)$. Specifically, ${\mu }_{(r)}=r!f_k((r+1)k+r)\mathrm{/}((q{p}^k{)}^{r+1})$.