Theorem: Negated Upper Index of Binomial Coefficient/Complex Numbers

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:
{{begin-eqn}}
{{eqn | l = \map \Gamma {\zeta - \omega + 1} \map \Gamma {1 - \paren {\zeta - \omega + 1} }
      | r = \dfrac \pi {\map \sin {\pi \paren {\zeta - \omega + 1} } }
      | c = Euler's Reflection Formula
}}
{{eqn | n = 1
      | ll= \leadsto
      | l = \map \Gamma {\zeta - \omega + 1}
      | r = \dfrac \pi {\map \sin {\pi \paren {\zeta - \omega + 1} } \map \Gamma {\omega - \zeta} }
      | c = 
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn | l = \map \Gamma {\zeta + 1} \map \Gamma {1 - \paren {\zeta + 1} }
      | r = \dfrac \pi {\map \sin {\pi \paren {\zeta + 1} } }
      | c = Euler's Reflection Formula
}}
{{eqn | n = 2
      | ll= \leadsto
      | l = \map \Gamma {\zeta + 1}
      | r = \dfrac \pi {\map \sin {\pi \paren {\zeta + 1} } \map \Gamma {-\zeta} }
      | c = 
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \dbinom z w
      | r = \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma {\omega + 1} \map \Gamma {\zeta - \omega + 1} }
      | c = {{Defof|Binomial Coefficient/Complex Numbers|Binomial Coefficient}}
}}
{{eqn | r = \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\map \Gamma {\zeta + 1} \paren {\map \sin {\pi \paren {\zeta - \omega + 1} } \map \Gamma {\omega - \zeta} } } {\map \Gamma {\omega + 1} \pi}
      | c = from $(1)$
}}
{{eqn | r = \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\pi \paren {\map \sin {\pi \paren {\zeta - \omega + 1} } \map \Gamma {\omega - \zeta} } } {\paren {\map \sin {\pi \paren {\zeta + 1} } \map \Gamma {-\zeta} } \map \Gamma {\omega + 1} \pi}
      | c = from $(2)$
}}
{{eqn | r = \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\map \sin {\pi \paren {\zeta - \omega + 1} } } {\map \sin {\pi \paren {\zeta + 1} } } \dfrac {\map \Gamma {\omega - \zeta} } {\map \Gamma {-\zeta} \map \Gamma {\omega + 1} }
      | c = rearranging
}}
{{eqn | r = \dfrac {\map \sin {\pi \paren {z - w + 1} } } {\map \sin {\pi \paren {z + 1} } } \ds \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\map \Gamma {\omega - \zeta} } {\map \Gamma {-\zeta} \map \Gamma {\omega + 1} }
      | c = Combination Theorem for Limits of Complex Functions
}}
{{eqn | r = \dfrac {\map \sin {\pi \paren {z - w + 1} } } {\map \sin {\pi \paren {z + 1} } } \ds \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\map \Gamma {\paren {\omega - \zeta - 1} + 1} } {\map \Gamma {\paren {\omega - \zeta - 1} + \omega + 1} \map \Gamma {\omega + 1} }
      | c = Combination Theorem for Limits of Complex Functions and rearrangement
}}
{{eqn | r = \dfrac {\map \sin {\pi \paren {z - w + 1} } } {\map \sin {\pi \paren {z + 1} } } \dbinom {w - z - 1} w
      | c = {{Defof|Binomial Coefficient/Complex Numbers|Binomial Coefficient}}
}}
{{end-eqn}}
Now we have:
{{begin-eqn}}
{{eqn | l = \map \sin {\pi \paren {z - w + 1} }
      | r = -\map \sin {-\pi \paren {z - w + 1} }
      | c = Sine Function is Odd
}}
{{eqn | r = -\map \sin {\pi \paren {-z + w - 1} }
      | c = 
}}
{{eqn | r = -\map \sin {\pi \paren {w - z - 1} }
      | c = 
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn | l = \map \sin {\pi \paren {z + 1} }
      | r = \map \sin {\pi z + \pi}
      | c = 
}}
{{eqn | r = -\map \sin {\pi z}
      | c = Sine of Angle plus Straight Angle
}}
{{end-eqn}}
Thus:
{{begin-eqn}}
{{eqn | l = \dfrac {\map \sin {\pi \paren {z - w + 1} } } {\map \sin {\pi \paren {z + 1} } }
      | r = \dfrac {-\map \sin {\pi \paren {w - z - 1} } } {-\map \sin {\pi z} }
      | c = 
}}
{{eqn | r = \dfrac {\map \sin {\pi \paren {w - z - 1} } } {\map \sin {\pi z} }
      | c = 
}}
{{end-eqn}}
and the result follows.
{{qed}}