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
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $42$ (Solution)