Orr's Theorem

$\begin{displaymath} (1-z)^{\alpha+\beta+\gamma-1/2}\,{}_2F_1(2\alpha,2\beta;2\gamma;z)=\sum a_nz^n, \end{displaymath}$

(1)

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

$\begin{displaymath} {}_2F_1(\alpha,\beta;\gamma;z)\,{}_2F_1(\gamma-\alpha+{\text... ... \sum_{(\gamma+{\textstyle{1\over 2}})_n/(\gamma+1)_n} a_nz^n. \end{displaymath}$

(2)

Furthermore, if

$\begin{displaymath} (1-z)^{\alpha+\beta-\gamma-1/2}\,{}_2F_1(2\alpha-1,2\beta;2\gamma-1;z)=\sum a_nz^n, \end{displaymath}$

(3)

then

$\begin{displaymath} {}_2F_1(\alpha,\beta;\gamma;z)\Gamma(\gamma-\alpha+{\textsty... ... = \sum_{(\gamma-{\textstyle{1\over 2}})_n/(\gamma)_n} a_nz^n, \end{displaymath}$

(4)

where $\Gamma(z)$ is the Gamma Function.