Dougall's Hypergeometric Theorem/Corollary 5

Corollary to Dougall's Hypergeometric Theorem

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} } $

Proof

Set $x = y = z = -n$ in Dougall's Hypergeometric Theorem

Before substitution:

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

\(=\)

\(\ds \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {z + n + 1} \map \Gamma {x + y + z + n + 1} } { \map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {y + z + n + 1} \map \Gamma {x + z + n + 1} }\)

After substitution:

\(\ds \map { {}_5 \operatorname F_4} { { {\dfrac n 2 + 1, n, -\paren {-n}, -\paren {-n}, -\paren {-n} } \atop {\dfrac n 2, \paren {-n} + n + 1, \paren {-n} + n + 1, \paren {-n} + n + 1} } \, \middle \vert \, 1}\)

\(=\)

\(\ds \dfrac {\map \Gamma {\paren {-n} + n + 1} \map \Gamma {\paren {-n} + n + 1} \map \Gamma {\paren {-n} + n + 1} \map \Gamma {\paren {-n} + \paren {-n} + \paren {-n} + n + 1} } { \map \Gamma {n + 1} \map \Gamma {\paren {-n} + \paren {-n} + n + 1} \map \Gamma {\paren {-n} + \paren {-n} + n + 1} \map \Gamma {\paren {-n} + \paren {-n} + n + 1} }\)

\(\ds \leadsto \ \ \)

\(\ds \map { {}_5 \operatorname F_4} { { {\dfrac n 2 + 1, n, n, n, n} \atop {\dfrac n 2,1, 1, 1} } \, \middle \vert \, 1}\)

\(=\)

\(\ds \dfrac {\map \Gamma 1 \map \Gamma 1 \map \Gamma 1 \map \Gamma {1 - 2n} } { \map \Gamma {1 + n} \paren {\map \Gamma {1 - n} }^3 }\)

\(\ds \leadsto \ \ \)

\(\ds \map { {}_5 \operatorname F_4} { { {\dfrac n 2 + 1, n, n, n, n} \atop {\dfrac n 2,1, 1, 1} } \, \middle \vert \, 1}\)

\(=\)

\(\ds \dfrac {\map \Gamma {1 - 2n} } { n \map \Gamma n \paren {\map \Gamma {1 - n} }^3 }\)

Definition of Gamma Function and $\map \Gamma {1} = 1$

\(\ds \leadsto \ \ \)

\(\ds \)

\(=\)

\(\ds \dfrac {\map \Gamma {1 - 2n} } { n \map \Gamma n \paren {\map \Gamma {1 - n} }^3 } \times \dfrac {\map \Gamma {2n} } {\map \Gamma {2n} } \times \dfrac {\paren {\map \Gamma n}^2 } {\paren {\map \Gamma n}^2 }\)

multiplying by $1$

\(\ds \leadsto \ \ \)

\(\ds \)

\(=\)

\(\ds \paren {\map \Gamma {1 - 2n} \map \Gamma {2n} } \paren {\dfrac 1 {\paren {\map \Gamma n }^3 \paren {\map \Gamma {1 - n} }^3} } \paren {\dfrac {\paren {\map \Gamma n}^2} {n \map \Gamma {2n} } }\)

rearranging

\(\ds \leadsto \ \ \)

\(\ds \)

\(=\)

\(\ds \paren {\dfrac \pi {\map \sin {2\pi n} } } \paren {\dfrac {\map \sin {\pi n} } \pi }^3 \paren {\dfrac {\paren {\map \Gamma n}^2} {n \map \Gamma {2n} } }\)

Euler's Reflection Formula

\(\ds \leadsto \ \ \)

\(\ds \)

\(=\)

\(\ds \paren {\dfrac \pi {2 \map \sin {\pi n} \map \cos {\pi n} } } \paren {\dfrac {\map \sin {\pi n} } \pi }^3 \paren {\dfrac {\paren {\map \Gamma n}^2} {n \map \Gamma {2n} } }\)

Double Angle Formula for Sine

\(\ds \leadsto \ \ \)

\(\ds \)

\(=\)

\(\ds \paren {\dfrac \pi {\map \sin {\pi n} \map \cos {\pi n} } } \paren {\dfrac {\map \sin {\pi n} } \pi }^3 \paren {\dfrac {\paren {\map \Gamma n}^2} {\map \Gamma {2n + 1} } }\)

Definition of Gamma Function

\(\ds \leadsto \ \ \)

\(\ds \)

\(=\)

\(\ds \dfrac {\paren {\map \Gamma n}^2 \map \sin {\pi n} \map \tan {\pi n} } {\pi^2 \map \Gamma {2n + 1} }\)

$\blacksquare$

Sources

• 1989: Bruce C. Berndt: Ramanujan's Notebooks: Part II: Chapter $\text {10}$. Hypergeometric Series: $\text I$