Dougall's Hypergeometric Theorem

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This proof is about Dougall's Theorem in the context of hypergeometric functions. For other uses, see Dougall's Theorem.

Theorem

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

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

Let $\map \Re {x + y + z + n + 1} > 0$.


Then:

$\ds \map { {}_5 \operatorname F_4} { { {\dfrac n 2 + 1, n, -x, -y, -z} \atop {\dfrac n 2, x + n + 1, y + n + 1, z + n + 1} } \, \middle \vert \, 1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {z + n + 1} \map \Gamma {x + y + z + n + 1} } {\map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {y + z + n + 1} \map \Gamma {x + z + n + 1} }$

where:

${}_5 \operatorname F_4$ 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.


Corollary 1

Let $\map \Re {x + y + 1} > 0$.

Then:

$\ds \map { {}_5 \operatorname F_4} { { {\dfrac n 2 + 1, n, n, -x, -y} \atop {\dfrac n 2, x + n + 1, y + n + 1, 1} } \, \middle \vert \, 1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {x + y + 1} } { \map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {y + 1} \map \Gamma {x + 1} } $


Corollary 2

Let $\map \Re {x + y + n} > 0$.

Then:

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


Corollary 3

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} } $


Corollary 4

Let $\map \Re {x - n + 1} > 0$.

Then:

$\ds \map { {}_5 \operatorname F_4} { { {\dfrac n 2 + 1, n, n, n, -x} \atop {\dfrac n 2, x + n + 1, 1, 1} } \, \middle \vert \, 1} = \dfrac {\map \sin {\pi n} \map \Gamma {x + n + 1} \map \Gamma {x - n + 1} } {\pi n \paren {\map \Gamma {x + 1} }^2 } $


Corollary 5

Let $\map \Re {n} < \dfrac 1 2$.

Then:

$\ds \map { {}_5 \operatorname F_4} { { {\dfrac n 2 + 1, n, n, n, n} \atop {\dfrac n 2, 1, 1, 1} } \, \middle \vert \, 1} = \dfrac {\paren {\map \Gamma n}^2 \map \sin {\pi n} \map \tan {\pi n} } {\pi^2 \map \Gamma {2n + 1} } $


Corollary 6

Let $\map \Re {n} < \dfrac 2 3$.

Then:

$\ds \map { {}_4 \operatorname F_3} { { {\dfrac n 2 + 1, n, n, n} \atop {\dfrac n 2, 1, 1} } \, \middle \vert \, -1} = \dfrac {\map \sin {\pi n} } {\pi n } $


Proof

By definition of the generalized hypergeometric function

$\ds \map { {}_5 \operatorname F_4} { { {\dfrac n 2 + 1, n, -x, -y, -z} \atop {\dfrac n 2, x + n + 1, y + n + 1, z + n + 1} } \, \middle \vert \, 1} = \ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {\dfrac n 2 + 1}^{\overline k} n^{\overline k} \paren {-x}^{\overline k} \paren {-y}^{\overline k} \paren {-z}^{\overline k} } {\paren {\dfrac n 2}^{\overline k} \paren {x + n + 1}^{\overline k} \paren {y + n + 1}^{\overline k} \paren {z + n + 1}^{\overline k} } \dfrac {1^k} {k!}$


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Also known as

Some sources refer to this theorem as Dougall's Theorem, but that is often used for the Dougall-Ramanujan Identity.


Examples

Example: $\map { {}_5 \operatorname F_4} {\dfrac 9 8, \dfrac 1 4, \dfrac 1 4, \dfrac 1 4, \dfrac 1 4; \dfrac 1 8, 1, 1, 1; 1}$

$1 + 9 \paren {\dfrac 1 4}^4 + 17 \paren {\dfrac {1 \times 5} {4 \times 8} }^4 + 25 \paren {\dfrac {1 \times 5 \times 9} {4 \times 8 \times 12} }^4 + \cdots = \dfrac {2 \sqrt 2 } { \sqrt \pi \paren {\map \Gamma {\dfrac 3 4} }^2 }$


Example: $\map { {}_4 \operatorname F_3} {\dfrac 5 4, \dfrac 1 2, \dfrac 1 2, \dfrac 1 2; \dfrac 1 4, 1, 1; -1}$

$1 - 5 \paren {\dfrac 1 2}^3 + 9 \paren {\dfrac {1 \times 3} {2 \times 4} }^3 - 13 \paren {\dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} }^3 + \cdots = \dfrac 2 \pi$


Also see


Source of Name

This entry was named for John Dougall.


Sources

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