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 $x$.

Proof
First, we observe:

Applying Pfaff's Transformation twice, we obtain:

Therefore, after two applications of Pfaff's Transformation, we have:
 * $\map F {a, b; c; x} = \paren {1 - x}^{c - a - b} \map F {c - a, c - b; c; x}$

Also see

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