Kummer's Quadratic Transformation

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

Let $z < 3 - 2 \sqrt 2$

Let $\size z < 1$

Then:
 * $\ds \map F {a, b; 1 + a - b; z} = \paren {1 - z}^{-a} \map F {\dfrac a 2, \dfrac {1 + a} 2 - b; 1 + a - b; \dfrac {-4 z} {\paren {1 - z}^2} }$

where:
 * $\map F {a, b; 1 + a - b; z}$ is the Gaussian hypergeometric function of $z$: $\ds \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} b^{\overline k} } {\paren {1 + a - b}^{\overline k} } \dfrac {z^k} {k!}$
 * $x^{\overline k}$ denotes the $k$th rising factorial power of $x$
 * $\map \Gamma {n + 1} = n!$ is the Gamma function.

Proof
On the, our $z$ variable transforms to $\dfrac {-4 z} {\paren {1 - z}^2}$, therefore:

By the Quadratic Formula, either $z > 3 + 2 \sqrt 2$ or $z < 3 - 2 \sqrt 2$.

Because $\size z < 1$, we have:
 * $\size z < 3 - 2 \sqrt 2$

This means our sum is analytic and can be expanded in powers in $z$ for $\size z < 3 - 2 \sqrt 2$

From the definition of Gaussian hypergeometric function, the can be written as:

The coefficient of $z^n$ on the is:

We now observe that the coefficients of $z^n$ on the are identical to the coefficients of $z^n$ on the.

Therefore:
 * $\ds \map F {a, b; 1 + a - b; z} = \paren {1 - z}^{-a} \map F {\dfrac a 2, \dfrac {1 + a} 2 - b; 1 + a - b; \dfrac {-4 z} {\paren {1 - z}^2} }$

Also see

 * Euler's Transformation
 * Pfaff's Transformation