Dougall's Hypergeometric Theorem/Examples/5F4(1.125,0.25,0.25,0.25,0.25;0.125,1,1,1;1)

Example of Use of Dougall's Hypergeometric Theorem

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

Proof
From Dougall's Hypergeometric Theorem:


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

where:
 * $\ds \map { {}_5 \operatorname F_4} { { {\dfrac n 2 + 1, n, -x, -y, -z} \atop {\dfrac n 2, x + n + 1, y + n + 1, z + n + 1} } \, \middle \vert \, 1}$ is the generalized hypergeometric function of $1$: $\ds \sum_{k \mathop = 0}^\infty \dfrac { \paren {\dfrac n 2 + 1}^{\overline k} n^{\overline k} \paren {-x}^{\overline k} \paren {-y}^{\overline k} \paren {-z}^{\overline k} } {\paren {\dfrac n 2}^{\overline k} \paren {x + n + 1}^{\overline k} \paren {y + n + 1}^{\overline k} \paren {z + n + 1}^{\overline k} } \dfrac {1^k} {k!}$
 * $x^{\overline k}$ denotes the $k$th rising factorial power of $x$
 * $\map \Gamma {n + 1} = n!$ is the Gamma function.

We have:

and:

Recall the Euler's Reflection Formula:
 * $\map \Gamma z \map \Gamma {1 - z} = \dfrac \pi {\map \sin {\pi z} }$

Therefore:

Substituting these results back into our equation above:

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