Definition:Binomial Coefficient/Complex Numbers

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Let $z, w \in \C$.

Then $\dbinom z w$ is defined as:

$\dbinom z w := \ds \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma {\omega + 1} \map \Gamma {\zeta - \omega + 1} }$

where $\Gamma$ denotes the Gamma function.

When $z$ is a negative integer and $w$ is not an integer, $\dbinom z w$ is infinite.

Also rendered as

Some sources give this as:

$\dbinom z w := \ds \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\zeta!} {\omega! \, \paren {\zeta - \omega}!}$

where $\zeta! := \map \Gamma {\zeta + 1}$.

This is unusual, however, as the factorial is usually defined only for positive integers.