Dixon's Hypergeometric Theorem/Examples

Examples of Use of Dixon's Hypergeometric Theorem‎

Example: $\map { {}_3 \operatorname F_2} {\dfrac 1 2, \dfrac 1 2, \dfrac 1 4; 1, \dfrac 5 4; 1}$

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

Example: $\map { {}_3 \operatorname F_2} {\dfrac 1 2, \dfrac 1 4, \dfrac 1 4; \dfrac 5 4, \dfrac 5 4; 1}$

$1 + \dfrac 1 {5^2} \paren {\dfrac 1 2} + \dfrac 1 {9^2} \paren {\dfrac {1 \times 3} {2 \times 4} } + \dfrac 1 {13^2} \paren {\dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} } + \cdots = \dfrac {\pi^{\frac 5 2} } {8 \sqrt 2 \paren {\map \Gamma {\dfrac 3 4} }^2}$

Example: $\map { {}_3 \operatorname F_2} {\dfrac 1 2, \dfrac 1 2, \dfrac 1 2; 1, 1; 1}$

$1 + \paren {\dfrac 1 2}^3 + \paren {\dfrac {1 \times 3} {2 \times 4} }^3 + \paren {\dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} }^3 + \cdots = \dfrac \pi {\paren {\map \Gamma {\dfrac 3 4} }^4 }$