Definition:Binomial Coefficient/Complex Numbers

Definition

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.

Sources

• 1977: Lyle Ramshaw: Binomial coefficients with non-integral lower index (Inf. Proc. Letters Vol. 6: pp. 223 – 226)
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): binomial coefficient: 1.
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $42$ (Solution)