Falling Factorial as Quotient of Factorials
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Theorem
Let $x \in \Z_{\ge 0}$ be a positive integer.
Then:
- $x^{\underline n} = \dfrac {x!} {\paren {x - n}!} = \dfrac {\map \Gamma {x + 1} } {\map \Gamma {x - n + 1} }$
where:
- $x^{\underline n}$ denotes the $n$th falling factorial power of $x$.
- $\map \Gamma x$ denotes the Gamma function of $x$.
Proof
\(\ds x^{\underline n}\) | \(=\) | \(\ds \prod_{j \mathop = 0}^{n - 1} \paren {x - j}\) | 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 |
$\blacksquare$
Also see
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials: $(21)$