Factors of Binomial Coefficient/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:

$\dbinom z w = \dfrac z w \dbinom {z - 1} {w - 1}$

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


Proof

\(\displaystyle \dbinom z w\) \(=\) \(\displaystyle \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\Gamma \left({\zeta + 1}\right)} {\Gamma \left({\omega + 1}\right) \Gamma \left({\zeta - \omega + 1}\right)}\) Definition of Binomial Coefficient
\(\displaystyle \) \(=\) \(\displaystyle \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\zeta \Gamma \left({\zeta}\right)} {\omega \Gamma \left({\omega}\right) \Gamma \left({\zeta - \omega + 1}\right)}\) Gamma Difference Equation
\(\displaystyle \) \(=\) \(\displaystyle \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac \zeta \omega \dfrac {\Gamma \left({\left({\zeta - 1}\right) + 1}\right)} {\omega \Gamma \left({\left({\omega - 1}\right) + 1}\right) \Gamma \left({\left({\zeta - 1}\right) - \left({\omega - 1}\right) + 1}\right)}\) rearrangement
\(\displaystyle \) \(=\) \(\displaystyle \dfrac z w \dbinom {z - 1} {w - 1}\) Definition of Binomial Coefficient

$\blacksquare$


Sources