Chu-Vandermonde Identity/Proof 2

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Theorem

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


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.

$\blacksquare$


Source of Name

This entry was named for Alexandre-Théophile Vandermonde and Chu Shih-Chieh.