Chu-Vandermonde Identity/Proof 2

Theorem
Let $r, s \in \R, n \in \Z$.

Then:


 * $\displaystyle \sum_k \binom r k \binom s {n-k} = \binom {r+s} n$

where $\displaystyle \binom r k$ is a binomial coefficient.

Proof 2
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:


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

throughout.