Dougall's Hypergeometric Theorem/Examples

Examples of Use of Dougall's Hypergeometric Theorem

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

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