Falling Factorial as Quotient of Factorials

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Theorem

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


Then:

$x^{\underline n} = \dfrac {x!} {\left({x - n}\right)!} = \dfrac {\Gamma \left({x + 1}\right)} {\Gamma \left({x - n + 1}\right)}$

where:

$x^{\underline n}$ denotes the $n$th falling factorial power of $x$.
$\Gamma \left({x}\right)$ denotes the Gamma function of $x$.


Proof

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

$\blacksquare$


Also see


Sources