Kummer's Hypergeometric Theorem/Proof 2

Proof
From Euler's Integral Representation of Hypergeometric Function, we have:
 * $\ds \map F {a, b; c; x} = \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - x t}^{- a} \rd t$

Where $a, b, c \in \C$.

and $\size x < 1$

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

Since Euler's Integral Representation only applies where $\size x < 1$, we will determine the limit of the integral as $x \to -1$.

By symmetry, we have:
 * $\ds \map F {n, -x; x + n + 1; -1} = \ds \map F {-x, n; x + n + 1; -1}$

Therefore:

We now apply a u-substitution: Let $u = t^2$

Substituting back into our equation, we have: