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:
 * $\dbinom z w = \dfrac {\sin \left({\pi \left({w - z - 1}\right)}\right)} {\sin \left({\pi z}\right)} \dbinom {w - z - 1} w$

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

Proof
By definition of Binomial Coefficient:
 * $\dbinom z w = \displaystyle \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\Gamma \left({\zeta + 1}\right)} {\Gamma \left({\omega + 1}\right) \Gamma \left({\zeta - \omega + 1}\right)}$

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

Thus:

and:

Hence:

Now we have:

and:

Thus:

and the result follows.