Gauss's Hypergeometric Theorem/Examples

Examples of Use of Gauss's Hypergeometric Theorem

Example: $\map F {1, 2; 4; 1}$

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

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

$1 + \dfrac 2 5 + \paren {\dfrac {2 \times 4} {5 \times 7} } + \paren {\dfrac {2 \times 4 \times 6} {5 \times 7 \times 9} } + \cdots = 3$

Example: $\map F {\dfrac 1 2, \dfrac 1 2; \dfrac 3 2; 1}$

$1 + \paren {\dfrac 1 {2^1 \times 3 \times 1!} } + \paren {\dfrac {1 \times 3} {2^2 \times 5 \times 2!} } + \paren {\dfrac {1 \times 3 \times 5} {2^3 \times 7 \times 3!} } + \cdots = \dfrac \pi 2$

Example: $\dfrac {10} 3 \map F {\dfrac 3 {10}, \dfrac 3 {10}; \dfrac {13} {10}; 1}$

$\paren {\dfrac 1 {10^{-1} \times 3 \times 0!} } + \paren {\dfrac 3 {10^0 \times 13 \times 1!} } + \paren {\dfrac {3 \times 13} {10^1 \times 23 \times 2!} } + \paren {\dfrac {3 \times 13 \times 23} {10^2 \times 33 \times 3!} } + \cdots = \dfrac {2 \pi} \phi$