Kummer's Hypergeometric Theorem/Proof 3

Proof
From Kummer's Quadratic Transformation, we have:


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

Let $z \to -1$ and we have:


 * $\ds \map F {a, b; 1 + a - b; -1} = 2^{-a} \map F {\dfrac a 2, \dfrac {1 + a} 2 - b; 1 + a - b; 1 }$

From Gauss's Hypergeometric Theorem, we have:


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

Therefore, the becomes:

Substituting $a = n$ and $b = -x$, we obtain: