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Orr's Theorem

If

\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.




© 1996-9 Eric W. Weisstein
1999-05-26