Chu-Vandermonde Identity/Proof 2

Proof
This is a special case of Gauss's Hypergeometric Theorem:


 * $\map { {}_2F_1} {a, b; c; 1} = \dfrac {\map \Gamma c \map \Gamma {c - a - b} } {\map \Gamma {c - a} \map \Gamma {c - b} }$

where:
 * ${}_2F_1$ is the hypergeometric series
 * $\map \Gamma {n + 1} = n!$ is the Gamma function.

One regains the Chu-Vandermonde Identity by taking $a = -n$ and applying Negated Upper Index of Binomial Coefficient:


 * $\dbinom n k = (-1)^k \dbinom {k - n - 1} k$

throughout.