Dixon's Hypergeometric Theorem/Corollary 1

Corollary to Dixon's Hypergeometric Theorem
Let $\map \Re {n} < \dfrac 2 3$.

Then:


 * $\ds 1 + \paren {\dfrac n {1!} }^3 + \paren {\dfrac {n \paren {n + 1} } {2!} }^3 + \paren {\dfrac {n \paren {n + 1} \paren {n + 2} } {3!} }^3 + \cdots = \dfrac {6 \map \sin {\dfrac {\pi n} 2} \map \sin {\pi n} \map {\Gamma^3} {\dfrac n 2 + 1} } {\pi^2 n^2 \paren {1 + 2 \map \cos {\pi n} } \map \Gamma {\dfrac {3 n} 2 + 1} } $

Proof
From Dixon's Hypergeometric Theorem:


 * $\ds \map { {}_3 \operatorname F_2} { { {n, -x, -y} \atop {x + n + 1, y + n + 1} } \, \middle \vert \, 1} = \dfrac {\map \Gamma {x + n + 1} \map \Gamma {y + n + 1} \map \Gamma {\dfrac n 2 + 1} \map \Gamma {x + y + \dfrac n 2 + 1} } { \map \Gamma {n + 1} \map \Gamma {x + y + n + 1} \map \Gamma {x + \dfrac n 2 + 1} \map \Gamma {y + \dfrac n 2 + 1} } $

where:
 * $\ds \map { {}_3 \operatorname F_2} { { {n, -x, -y} \atop {x + n + 1, y + n + 1} } \, \middle \vert \, 1}$ is the generalized hypergeometric function of $1$: $\ds \sum_{k \mathop = 0}^\infty \dfrac { n^{\overline k} \paren {-x}^{\overline k} \paren {-y}^{\overline k} } { \paren {x + n + 1}^{\overline k} \paren {y + n+ 1}^{\overline k} } \dfrac {1^k} {k!}$
 * $x^{\overline k}$ denotes the $k$th rising factorial power of $x$
 * $\map \Gamma {n + 1} = n!$ is the Gamma function.

Set $x = y = -n$:

and:

Note the $\map \sin {\dfrac {3\pi n} 2 }$ in the denominator which is why $\map \Re {n} < \dfrac 2 3$.

Also known as
The corollary above (from Srinivasa Ramanujan) is sometimes referred to as Morley's Formula after mathematician Frank Morley.

However, it is usually shown in the following simplified format:

The corollary above is equivalent to this statement as can be seen below: