Dougall's Hypergeometric Theorem/Corollary 3

Corollary to Dougall's Hypergeometric Theorem
Let $\map \Re {2x + 2y + n + 2} > 0$.

Let $n \notin \Z_{\lt 0}$

Then:


 * $\ds \map { {}_4 \operatorname F_3} { { {\dfrac n 2 + 1, n, -x, -y} \atop {\dfrac n 2, x + n + 1, y + n + 1} } \, \middle \vert \, -1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} } {\map \Gamma {n + 1} \map \Gamma {x + y + n + 1} } $

Proof
Two lemmata:

Lemma
Let $z \to \infty$ in Dougall's Hypergeometric Theorem

We have: