Binomial Coefficient as Infinite Product
Theorem
| \(\ds \prod_{n \mathop = 1}^\infty \frac {\paren {n + b} } n \frac {\paren {n + a - b} } {\paren {n + a} }\) | \(=\) | \(\ds \frac {\paren {1 + b} } 1 \frac {\paren {1 + a - b} } {\paren {1 + a} } \times \frac {\paren {2 + b} } 2 \frac {\paren {2 + a - b} } {\paren {2 + a} } \times \cdots\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dbinom a b\) |
Proof
| \(\ds \prod_{n \mathop = 1}^\infty \frac {\paren {n + b} } n \frac {\paren {n + a - b} } {\paren {n + a} }\) | \(=\) | \(\ds \frac {\map \Gamma 1 \map \Gamma {a + 1} } {\map \Gamma {b + 1} \map \Gamma {a - b + 1} }\) | Infinite Product of Product of Sequence of n plus alpha over Sequence of n plus beta | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {a!} {b! \paren {a - b}!}\) | Gamma Function Extends Factorial | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dbinom a b\) | Definition of Binomial Coefficient |
$\blacksquare$