Pfaff's Transformation

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

Let $\size x < \dfrac 1 2$

Let $\map \Re c > \map \Re b > 0$.

Then:
 * $\ds \map F {a, b; c; x} = \paren {1 - x}^{-a} \map F {a, c - b; c; \dfrac x {x - 1} }$

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 $x$
 * $\map \Gamma {n + 1} = n!$ is the Gamma function.

Proof
From Euler's Integral Representation of Hypergeometric Function, we have:


 * $\ds \map F {a, b; c; x} = \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - xt}^{- a} \rd t$

Letting $s = \paren {1 - t}$, we now have:

Letting $\size x < \dfrac 1 2$ and expanding the product of $\paren {1 - \dfrac {xs} {x - 1} }^{-a}$:

Therefore:

Therefore:
 * $\ds \map F {a, b; c; x} = \paren {1 - x}^{-a} \map F {a, c - b; c; \dfrac x {x - 1} }$

Also see

 * Euler's Integral Representation of Hypergeometric Function
 * Euler's Transformation
 * Kummer's Quadratic Transformation