Derivative of Gaussian Hypergeometric Function

Theorem

 * $\map {\dfrac \d {\d x} } {\map F {a, b; c; x} } = \dfrac {a b} c \map F {a + 1, b + 1; c + 1; x} $

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

Also see

 * Gauss's Hypergeometric Theorem