Euler's Transformation

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

Let $\size x < 1$

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

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

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

Proof
First, we observe:

Next, from Pfaff's Transformation, we have:


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

Applying Pfaff's Transformation twice, we obtain:

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

Also see

 * Euler's Integral Representation of Hypergeometric Function
 * Pfaff's Transformation