Rising Factorial as Quotient of Factorials

Theorem
Let $x \in \Z_{\ge 0}$ be a positive integer.

Then:
 * $x^{\overline n} = \dfrac {\paren {x + n - 1}!} {\paren {x - 1}!} = \dfrac {\map \Gamma {x + n} } {\map \Gamma x}$

where:
 * $x^{\overline n}$ denotes the $n$th rising factorial power of $x$
 * $\map \Gamma x$ denotes the Gamma function of $x$.

Also see

 * Falling Factorial as Quotient of Factorials