Pascal's Rule/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 $w$ an integer:
- $\dbinom z {w - 1} + \dbinom z w = \dbinom {z + 1} w$
where $\dbinom z w$ is a binomial coefficient.
Proof
| \(\ds \binom z {w - 1} + \binom z w\) | \(=\) | \(\ds \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma \omega \map \Gamma {\zeta - \omega + 2} } + \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma {\omega + 1} \map \Gamma {\zeta - \omega + 1} }\) | Definition of Binomial Coefficient | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \paren {\dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma \omega \map \Gamma {\zeta - \omega + 2} } + \dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma {\omega + 1} \map \Gamma {\zeta - \omega + 1} } }\) | Sum Rule for Limits of Complex Functions | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \paren {\dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma \omega \paren {\zeta - \omega + 1} \map \Gamma {\zeta - \omega + 1} } + \dfrac {\map \Gamma {\zeta + 1} } {\omega \map \Gamma \omega \map \Gamma {\zeta - \omega + 1} } }\) | Gamma Difference Equation | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \paren {\dfrac {\omega \map \Gamma {\zeta + 1} + \paren {\zeta - \omega + 1} \map \Gamma {\zeta + 1} } {\omega \map \Gamma \omega \paren {\zeta - \omega + 1} \map \Gamma {\zeta - \omega + 1} } }\) | common demonimator | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \paren {\dfrac {\omega \map \Gamma {\zeta + 1} + \paren {\zeta - \omega + 1} \map \Gamma {\zeta + 1} } {\map \Gamma {\omega + 1} \map \Gamma {\zeta - \omega + 2} } }\) | Gamma Difference Equation | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \paren {\dfrac {\map \Gamma {\zeta + 1} \paren {\zeta - \omega + 1 + \omega} } {\map \Gamma {\omega + 1} \map \Gamma {\zeta - \omega + 2} } }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \paren {\dfrac {\map \Gamma {\zeta + 1} \paren {\zeta + 1} } {\map \Gamma {\omega + 1} \map \Gamma {\zeta - \omega + 2} } }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \paren {\dfrac {\map \Gamma {\zeta + 2} } {\map \Gamma {\omega + 1} \map \Gamma {\zeta - \omega + 2} } }\) | Gamma Difference Equation | |||||||||||
| \(\ds \) | \(=\) | \(\ds \binom {z + 1} w\) | Definition of Binomial Coefficient |
$\blacksquare$
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)