Factors 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 $w$ an integer:
- $\dbinom z w = \dfrac z w \dbinom {z - 1} {w - 1}$
where $\dbinom z w$ is a binomial coefficient.
Proof
\(\ds \dbinom z w\) | \(=\) | \(\ds \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\Gamma \left({\zeta + 1}\right)} {\Gamma \left({\omega + 1}\right) \Gamma \left({\zeta - \omega + 1}\right)}\) | Definition of Binomial Coefficient | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\zeta \Gamma \left({\zeta}\right)} {\omega \Gamma \left({\omega}\right) \Gamma \left({\zeta - \omega + 1}\right)}\) | Gamma Difference Equation | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac \zeta \omega \dfrac {\Gamma \left({\left({\zeta - 1}\right) + 1}\right)} {\omega \Gamma \left({\left({\omega - 1}\right) + 1}\right) \Gamma \left({\left({\zeta - 1}\right) - \left({\omega - 1}\right) + 1}\right)}\) | rearrangement | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac z w \dbinom {z - 1} {w - 1}\) | 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)