Falling Factorial as Quotient of Factorials

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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

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

$\blacksquare$


Also see


Sources