Kummer's Hypergeometric Theorem

Theorem
Let $x, n \in \C$.

Let $n \notin \Z_{< 0}$.

Let $\map \Re {x + 1} > 0$.

Then:
 * $\map F {n, -x; x + n + 1; -1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {\dfrac n 2 + 1} } {\map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {n + 1} }$

where:
 * $\map F {n, -x; x + n + 1; -1}$ is the Gaussian hypergeometric function of $-1$
 * $\map \Gamma {n + 1} = n!$ is the Gamma function.

Also see

 * Dixon's Hypergeometric Theorem
 * Gauss's Hypergeometric Theorem
 * Properties of Generalized Hypergeometric Function