Negated Upper Index of Binomial Coefficient/Complex Numbers

Theorem
For all $z, w \in \C$ such that it is not the case that $z$ is a negative integer and $t, w$ integers:
 * $\dbinom z w = \dfrac {\map \sin {\pi \paren {w - z - 1} } } {\map \sin {\pi z} } \dbinom {w - z - 1} w$

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

Proof
By definition of Binomial Coefficient:
 * $\dbinom z w = \ds \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma {\omega + 1} \map \Gamma {\zeta - \omega + 1} }$

Euler's Reflection Formula gives:
 * $\forall z \notin \Z: \map \Gamma z \map \Gamma {1 - z} = \dfrac \pi {\map \sin {\pi z} }$

Thus:

and:

Hence:

Now we have:

and:

Thus:

and the result follows.