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:


 * $\displaystyle {}_2F_1(a,b;c;1) = \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}$

$\displaystyle {}_2F_1$ is the Hypergeometric Series and $\Gamma(n+1)=n!$ is the Gamma function.

One regains the Chu-Vandermonde identity by taking $a = -n$ and applying the identity


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

liberally.