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 $x$.

$\ds x^{\overline n}$

$=$

$\ds \prod_{j \mathop = 0}^{n - 1} \paren {x + j}$

Definition of Rising Factorial

$\ds $

$=$

$\ds x \paren {x + 1} \paren {x + 2} \dotsm \paren {x + n - 1}$

$\ds $

$=$

$\ds \dfrac {\paren {x + n - 1}!} {\paren {x - 1}!}$

Definition of Factorial

$\ds $

$=$

$\ds \dfrac {\map \Gamma {x + n} } {\map \Gamma x}$

$\blacksquare$

1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials: $(21)$