Gauss's Hypergeometric Theorem/Proof 1

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: