Gauss's Hypergeometric Theorem

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

Let $\map \Re {c - a - b} > 0$.

Then:
 * $\map { {}_2F_1} {a, b; c; 1} = \dfrac {\map \Gamma c \map \Gamma {c - a - b} } {\map \Gamma {c - a} \map \Gamma {c - b} }$

where:
 * $\map { {}_2F_1} {a, b; c; 1}$ is the hypergeometric series: $\ds \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} b^{\overline k} } { c^{\overline k} } \dfrac {1^k} {k!}$
 * $x^{\overline k}$ denotes the $k$th rising factorial power of $x$
 * $\map \Gamma {n + 1} = n!$ is the Gamma function.

Proof
Let $x, y, n \in \C$ be complex numbers such that $\map \Re {x + y + n + 1} > 0$.

Let $u \in \C$ be a complex number such that $\cmod u < 1$.

Expanding the product of $\paren {1 + u}^{y + n}$ and $\paren {\dfrac {1 + u} u}^x$:

The coefficient $a_n$ of $u^n$ of the product $\dfrac {\paren {1 + u}^{x + y + n} } {u^x}$ above can be determined by setting $k = k + n$ in the series with $\dbinom {y + n} k$:

Now expand $\paren {1 + u}^{x + y + n}$ and divide by $u^x$:

The coefficient $a_n$ of $u^n$ of the product $\dfrac {\paren {1 + u}^{x + y + n} } {u^x}$ above is:

Equating coefficients gives us:

Therefore:

Letting $a = -x$, $b = -y$ and $c = n + 1$, we obtain: