Definition:Binomial Coefficient/Complex Numbers

Definition
Let $z, w \in \C$.

Then $\dbinom z w$ is defined as:
 * $\dbinom z w := \displaystyle \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\Gamma \left({\zeta + 1}\right)} {\Gamma \left({\omega + 1}\right) \Gamma \left({\zeta - \omega + 1}\right)}$

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:


 * $\dbinom z w := \displaystyle \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\zeta!} {\omega! \, \left({\zeta - \omega}\right)!}$

where $\zeta! := \Gamma \left({\zeta + 1}\right)$.

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