Binomial Coefficient expressed using Beta Function
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Theorem
Let $\dbinom r k$ denote a binomial coefficient.
Then:
- $\dbinom r k = \dfrac 1 {\paren {r + 1} \map B {k + 1, r - k + 1} }$
Proof
\(\ds \dbinom r k\) | \(=\) | \(\ds \dfrac {r!} {k! \, \paren {r - k}!}\) | Definition 1 of Binomial Coefficient | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\map \Gamma {r + 1} } {\map \Gamma {k + 1} \map \Gamma {r - k + 1} }\) | Gamma Function Extends Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\map \Gamma {r + 2} } {r + 1} \dfrac 1 {\map \Gamma {k + 1} \map \Gamma {r - k + 1} }\) | Gamma Difference Equation | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {r + 1} \dfrac {\map \Gamma {r + 2} } {\map \Gamma {k + 1} \map \Gamma {r - k + 1} }\) | rearranging | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {r + 1} \dfrac 1 {\map B {k + 1, r - k + 1} }\) | Definition 3 of Beta Function |
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $42$