Dougall-Ramanujan Identity

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Theorem

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

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


Then:

\(\ds \map { {}_7 \operatorname F_6} { { {n, 1 + \dfrac n 2, -x, -y, -z, -u, x + y + z + u + 2n + 1} \atop {\dfrac n 2, x + n + 1, y + n + 1, z + n + 1, u + n + 1, -x - y - z - u - n} } \, \middle \vert \, 1}\) \(=\) \(\ds \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {z + n + 1} \map \Gamma {x + y + z + n + 1} } {\map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {y + z + n + 1} \map \Gamma {x + z + n + 1} }\)
\(\ds \) \(\) \(\, \ds \times \, \) \(\ds \dfrac {\map \Gamma {u + n + 1} \map \Gamma {x + z + u + n + 1} \map \Gamma {y + z + u + n + 1} \map \Gamma {x + y + u + n + 1} } {\map \Gamma {x + u + n + 1} \map \Gamma {z + u + n + 1} \map \Gamma {y + u + n + 1} \map \Gamma {x + y + z + u + n + 1} }\)

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 $1$:

$\ds \map { {}_7 \operatorname F_6} { { {n, 1 + \dfrac n 2, -x, -y, -z, -u, x + y + z + u + 2n + 1} \atop {\dfrac n 2, x + n + 1, y + n + 1, z + n + 1, u + n + 1, -x - y - z - u - n} } \, \middle \vert \, 1} = \sum_{k \mathop = 0}^\infty \dfrac { n^{\overline k} \paren {1 + \dfrac n 2}^{\overline k} \paren {-x}^{\overline k} \paren {-y}^{\overline k} \paren {-z}^{\overline k} \paren {-u}^{\overline k} \paren {x + y + z + u + 2n + 1}^{\overline k} } { \paren {\dfrac n 2}^{\overline k} \paren {x + n + 1}^{\overline k} \paren {y + n + 1}^{\overline k} \paren {z + n + 1}^{\overline k} \paren {u + n + 1}^{\overline k} \paren {-x - y - z - u - n}^{\overline k} } \dfrac {1^k} {k!}$


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Also known as

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


Also see


Source of Name

This entry was named for John Dougall and Srinivasa Ramanujan.


Sources

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