Kummer's Hypergeometric Theorem/Examples

Examples of Use of Kummer's Hypergeometric Theorem

Example: $\map F {\dfrac 1 2, \dfrac 1 2; 1; -1}$

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

Example: $\map F {\dfrac 1 3, \dfrac 1 3; 1; -1}$

$1 - \paren {\dfrac 1 3}^2 + \paren {\dfrac {1 \times 4} {3 \times 6} }^2 - \paren {\dfrac {1 \times 4 \times 7} {3 \times 6 \times 9} }^2 + \cdots = \dfrac {\pi} {\paren {\map \Gamma {\dfrac 5 6} }^2 \map \Gamma {\dfrac 1 3} }$

Example: $\map F {\dfrac 2 5, \dfrac 1 {10}; \dfrac {13} {10}; -1}$

$1 - \paren {\dfrac {\paren {2} \paren {1} } {\paren {5} \paren {13} } } + \paren {\dfrac {\paren {2 \times 7} \paren {1 \times 11} } {\paren {5 \times 10} \paren {13 \times 23} } } - \paren {\dfrac {\paren {2 \times 7 \times 12} \paren {1 \times 11 \times 21} } {\paren {5 \times 10 \times 15} \paren {13 \times 23 \times 33} } } + \cdots = \dfrac {1944^{\frac 1 5} \pi^{\frac 3 2} } {\phi \map \Gamma {\dfrac 1 {10} } \paren {\map \Gamma {\dfrac 7 {10} } }^2 }$