Product of r Choose m with m Choose k/Complex Numbers

From ProofWiki
Jump to navigation Jump to search

Theorem

For all $z, w \in \C$ such that it is not the case that $z$ is a negative integer and $t, w$ integers:

$\dbinom z t \dbinom t w = \dbinom z w \dbinom {z - w} {t - w}$

where $\dbinom z w$ is a binomial coefficient.


Proof

\(\ds \dbinom z t \dbinom t w\) \(=\) \(\ds \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to t} \dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma {\tau + 1} \map \Gamma {\zeta - \tau + 1} } \lim_{\tau \mathop \to t} \lim_{\omega \mathop \to w} \dfrac {\map \Gamma {\tau + 1} } {\map \Gamma {\omega + 1} \map \Gamma {\tau - \omega + 1} }\) Definition of Binomial Coefficient
\(\ds \) \(=\) \(\ds \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to t} \paren {\dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma {\tau + 1} \map \Gamma {\zeta - \tau + 1} } \dfrac {\map \Gamma {\tau + 1} } {\map \Gamma {\omega + 1} \map \Gamma {\tau - \omega + 1} } }\) Product Rule for Limits of Complex Functions
\(\ds \) \(=\) \(\ds \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to t} \paren {\dfrac {\map \Gamma {\zeta + 1} } {\map \Gamma {\zeta - \tau + 1} \map \Gamma {\omega + 1} \map \Gamma {\tau - \omega + 1} } }\)
\(\ds \) \(=\) \(\ds \lim_{\zeta \mathop \to z} \lim_{\tau \mathop \to t} \paren {\dfrac {\map \Gamma {\zeta + 1} \map \Gamma {\zeta - \omega + 1} } {\map \Gamma {\omega + 1} \map \Gamma {\zeta - \omega + 1} \map \Gamma {\zeta - \tau + 1} \map \Gamma {\tau - \omega + 1} } }\)
\(\ds \) \(=\) \(\ds \binom z w \binom {z - w} {t - w}\) Definition of Binomial Coefficient

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Product_of_r_Choose_m_with_m_Choose_k/Complex_Numbers&oldid=499978"