Chu-Vandermonde Identity/Proof 2

Proof
The Chu-Vandermonde Identity is a special case of Gauss's Hypergeometric Theorem:


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

when:
 * $\map \Re {c - a - b} \gt 0$

where:
 * $\map { {}_2F_1} {a, b; c; 1}$ is the hypergeometric series: $\ds \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} b^{\overline k} } { c^{\overline k} } \dfrac {1^k} {k!}$
 * $x^{\overline k}$ denotes the $k$th rising factorial power of $x$
 * $\map \Gamma {n + 1} = n!$ is the Gamma function.

Starting on the :

Moving to the, we let $a = -n$:

Finally, setting the equal to the, we see the Chu-Vandermonde Identity:


 * $\ds \dfrac 1 { c^{\overline n} } \sum_{k \mathop = 0}^n \dbinom n k \paren {1 - b - k}^{\overline k} \paren {c - 1 + k}^{\overline {n - k} }   = \dfrac {\paren {c - b}^{\overline n} } {c^{\overline n} }$