Rising Factorial as Quotient of Factorials

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

Then:
 * $x^{\overline n} = \dfrac {\left({x + n - 1}\right)!} {\left({x - 1}\right)!} = \dfrac {\Gamma \left({x + n}\right)}{\Gamma \left({x}\right)}$

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

Also see

 * Falling Factorial as Quotient of Factorials