# 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 $x$.

## Proof

 $\ds x^{\underline n}$ $=$ $\ds \prod_{j \mathop = 0}^{n - 1} \paren {x - j}$ Definition of Falling Factorial $\ds$ $=$ $\ds x \paren {x - 1} \paren {x - 2} \dotsm \paren {x - n + 1}$ $\ds$ $=$ $\ds \dfrac {x!} {\paren {x - n}!}$ Definition of Factorial $\ds$ $=$ $\ds \dfrac {\map \Gamma {x + 1} } {\map \Gamma {x - n + 1} }$ Definition of Gamma Function

$\blacksquare$