Rising Factorial as Quotient of Factorials

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Theorem

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


Then:

$x^{\overline n} = \dfrac {\paren {x + n - 1}!} {\paren {x - 1}!} = \dfrac {\map \Gamma {x + n} } {\map \Gamma x}$

where:

$x^{\overline n}$ denotes the $n$th rising factorial power of $x$
$\map \Gamma x$ denotes the Gamma function of $x$.


Proof

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

$\blacksquare$


Also see


Sources