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:

$\dbinom z w = \ds \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma {\omega + 1} \map \Gamma {\zeta - \omega + 1} }$
$\forall z \notin \Z: \map \Gamma z \map \Gamma {1 - z} = \dfrac \pi {\map \sin {\pi z} }$

Thus:

 $\ds \map \Gamma {\zeta - \omega + 1} \map \Gamma {1 - \paren {\zeta - \omega + 1} }$ $=$ $\ds \dfrac \pi {\map \sin {\pi \paren {\zeta - \omega + 1} } }$ Euler's Reflection Formula $\text {(1)}: \quad$ $\ds \leadsto \ \$ $\ds \map \Gamma {\zeta - \omega + 1}$ $=$ $\ds \dfrac \pi {\map \sin {\pi \paren {\zeta - \omega + 1} } \map \Gamma {\omega - \zeta} }$

and:

 $\ds \map \Gamma {\zeta + 1} \map \Gamma {1 - \paren {\zeta + 1} }$ $=$ $\ds \dfrac \pi {\map \sin {\pi \paren {\zeta + 1} } }$ Euler's Reflection Formula $\text {(2)}: \quad$ $\ds \leadsto \ \$ $\ds \map \Gamma {\zeta + 1}$ $=$ $\ds \dfrac \pi {\map \sin {\pi \paren {\zeta + 1} } \map \Gamma {-\zeta} }$

Hence:

 $\ds \dbinom z w$ $=$ $\ds \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma {\omega + 1} \map \Gamma {\zeta - \omega + 1} }$ Definition of Binomial Coefficient $\ds$ $=$ $\ds \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\map \Gamma {\zeta + 1} \paren {\map \sin {\pi \paren {\zeta - \omega + 1} } \map \Gamma {\omega - \zeta} } } {\map \Gamma {\omega + 1} \pi}$ from $(1)$ $\ds$ $=$ $\ds \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\pi \paren {\map \sin {\pi \paren {\zeta - \omega + 1} } \map \Gamma {\omega - \zeta} } } {\paren {\map \sin {\pi \paren {\zeta + 1} } \map \Gamma {-\zeta} } \map \Gamma {\omega + 1} \pi}$ from $(2)$ $\ds$ $=$ $\ds \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\map \sin {\pi \paren {\zeta - \omega + 1} } } {\map \sin {\pi \paren {\zeta + 1} } } \dfrac {\map \Gamma {\omega - \zeta} } {\map \Gamma {-\zeta} \map \Gamma {\omega + 1} }$ rearranging $\ds$ $=$ $\ds \dfrac {\map \sin {\pi \paren {z - w + 1} } } {\map \sin {\pi \paren {z + 1} } } \ds \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\map \Gamma {\omega - \zeta} } {\map \Gamma {-\zeta} \map \Gamma {\omega + 1} }$ Combination Theorem for Limits of Complex Functions $\ds$ $=$ $\ds \dfrac {\map \sin {\pi \paren {z - w + 1} } } {\map \sin {\pi \paren {z + 1} } } \ds \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to w} \dfrac {\map \Gamma {\paren {\omega - \zeta - 1} + 1} } {\map \Gamma {\paren {\omega - \zeta - 1} + \omega + 1} \map \Gamma {\omega + 1} }$ Combination Theorem for Limits of Complex Functions and rearrangement $\ds$ $=$ $\ds \dfrac {\map \sin {\pi \paren {z - w + 1} } } {\map \sin {\pi \paren {z + 1} } } \dbinom {w - z - 1} w$ Definition of Binomial Coefficient

Now we have:

 $\ds \map \sin {\pi \paren {z - w + 1} }$ $=$ $\ds -\map \sin {-\pi \paren {z - w + 1} }$ Sine Function is Odd $\ds$ $=$ $\ds -\map \sin {\pi \paren {-z + w - 1} }$ $\ds$ $=$ $\ds -\map \sin {\pi \paren {w - z - 1} }$

and:

 $\ds \map \sin {\pi \paren {z + 1} }$ $=$ $\ds \map \sin {\pi z + \pi}$ $\ds$ $=$ $\ds -\map \sin {\pi z}$ Sine of Angle plus Straight Angle

Thus:

 $\ds \dfrac {\map \sin {\pi \paren {z - w + 1} } } {\map \sin {\pi \paren {z + 1} } }$ $=$ $\ds \dfrac {-\map \sin {\pi \paren {w - z - 1} } } {-\map \sin {\pi z} }$ $\ds$ $=$ $\ds \dfrac {\map \sin {\pi \paren {w - z - 1} } } {\map \sin {\pi z} }$

and the result follows.

$\blacksquare$