Symmetry Rule for Binomial Coefficients/Complex Numbers

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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

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