\[F\left({a,b\atop a-b+1};z\right)=\left(1+\sqrt{z}\right)^{-2a}F\left({a,a-b+%
\tfrac{1}{2}\atop 2a-2b+1};\frac{4\sqrt{z}}{(1+\sqrt{z})^{2}}\right),\]