Falling Factorial as Quotient of Factorials

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

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

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

Also see

 * Rising Factorial as Quotient of Factorials