# Rising Factorial as Quotient of Factorials

## 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

$\blacksquare$