Negated Upper Index 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 $t, w$ 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:

\(\ds \map \Gamma {\zeta - \omega + 1} \map \Gamma {1 - \paren {\zeta - \omega + 1} }\) \(=\) \(\ds \dfrac \pi {\map \sin {\pi \paren {\zeta - \omega + 1} } }\) 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.

$\blacksquare$


Sources

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