Falling Factorial as Quotient of Factorials

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 $x$.

$\ds x^{\underline n}$

$=$

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

Definition of Falling Factorial

$\ds $

$=$

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

$\ds $

$=$

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

Definition of Factorial

$\ds $

$=$

$\ds \dfrac {\map \Gamma {x + 1} } {\map \Gamma {x - n + 1} }$

Definition of Gamma Function

$\blacksquare$

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