Chu-Vandermonde Identity/Proof 2

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


 * ${}_2F_1 \left({a, b; c; 1}\right) = \dfrac{\Gamma \left({c}\right) \Gamma \left({c - a - b}\right)} {\Gamma \left({c - a}\right) \Gamma \left({c - b}\right)}$

where:
 * ${}_2F_1$ is the hypergeometric series
 * $\Gamma \left({n + 1}\right) = n!$ 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.