Symmetry Rule for Binomial Coefficients/Complex Numbers

Theorem

For all $z, w \in \C$ such that it is not the case that $z$ is a negative integer and $w$ an integer:

$\dbinom z w = \dbinom z {z - w}$

where $\dbinom z w$ is a binomial coefficient.

Proof

From the definition of the binomial coefficient:

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

where $\Gamma$ denotes the Gamma function.

$\ds \dbinom z w$

$=$

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

$\ds $

$=$

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

$\ds $

$=$

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

$\ds $

$=$

$\ds \dbinom z {z - w}$

$\blacksquare$

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 $42$ (Solution)