Dougall-Ramanujan Identity

Theorem
Let $x, y, z, u, n \in \C$.

Let at least one of $x, y, z, u, n, -x - y - z - u - 2n - 1 \in \Z_{>0}$.

Then:
 * $\ds \map { {}_7 \operatorname F_6} { { {n, 1 + \dfrac n 2, -x, -y, -z, -u, x - y + z + u + 2n + 1} \atop {\dfrac n 2, x + n + 1, y + n + 1, z + n + 1, u + n + 1, -x - y - z - u - n} } \, \middle \vert \, 1} = \dfrac 1 { \map \Gamma {n + 1} \map \Gamma {x + y + z + u + n + 1} } \prod_{x, y, z, u } \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + z + u + n + 1} } {\map \Gamma {z + u + n + 1} }$

where:
 * ${}_7 \operatorname F_6$ is the generalized hypergeometric function
 * $x^{\overline k}$ denotes the $k$th rising factorial power of $x$
 * $\map \Gamma {n + 1} = n!$ is the Gamma function.

Proof
By definition of generalized hypergeometric function of $1$:
 * $\ds \map { {}_7 \operatorname F_6} { { {n, 1 + \dfrac n 2, -x, -y, -z, -u, x - y + z + u + 2n + 1} \atop {\dfrac n 2, x + n + 1, y + n + 1, z + n + 1, u + n + 1, -x - y - z - u - n} } \, \middle \vert \, 1} = \sum_{k \mathop = 0}^\infty \dfrac { n^{\overline k} \paren {1 + \dfrac n 2}^{\overline k} \paren {-x}^{\overline k} \paren {-y}^{\overline k} \paren {-z}^{\overline k} \paren {-u}^{\overline k} \paren {x - y + z + u + 2n + 1}^{\overline k} } { \paren {\dfrac n 2}^{\overline k} \paren {x + n + 1}^{\overline k} \paren {y + n + 1}^{\overline k} \paren {z + n + 1}^{\overline k} \paren {u + n + 1}^{\overline k} \paren {-x - y - z - u - n}^{\overline k} } \dfrac {1^k} {k!}$

Also known as
Some sources refer to this theorem as Dougall's Theorem.

Also see

 * Dixon's Hypergeometric Theorem
 * Dougall's Hypergeometric Theorem
 * Gauss's Hypergeometric Theorem
 * Kummer's Hypergeometric Theorem