Cayley's Hypergeometric Function Theorem

If

$$(1-z)^{a+b-c}\,{}_2F_1(2a,2b;2c;z)=\sum_{n=0}^\infty a_nz^n,$$

then

$${}_2F_1(a,b;c+{\textstyle{1\over 2}};z)\,{}_2F_1(c-a,c-b;c{\... ...m_{n=0}^\infty {(c)_n\over (c+{\textstyle{1\over 2}})} a_nz^n,$$

where ${}_2F_1(a,b;c;z)$ is a Hypergeometric Function.

See also Hypergeometric Function