Gauss's Hypergeometric Theorem

Theorem
Let $a, b, c \in \C$.

Let $c \notin \Z_{\le 0}$.

Let $\map \Re {c - a - b} > 0$.

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

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

Also see

 * Dixon's Hypergeometric Theorem
 * Dougall's Hypergeometric Theorem
 * Kummer's Hypergeometric Theorem