\[F\left({a,b\atop a-b+1};z\right)=(1-z)^{-a}\frac{\Gamma\left(a-b+1\right)%
\Gamma\left(\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)%
\Gamma\left(\tfrac{1}{2}a-b+1\right)}F\left({\tfrac{1}{2}a,\tfrac{1}{2}a-b+%
\tfrac{1}{2}\atop\tfrac{1}{2}};\left(\frac{z+1}{z-1}\right)^{2}\right)+(1+z)(1%
-z)^{-a-1}\frac{\Gamma\left(a-b+1\right)\Gamma\left(-\tfrac{1}{2}\right)}{%
\Gamma\left(\tfrac{1}{2}a\right)\Gamma\left(\tfrac{1}{2}a-b+\tfrac{1}{2}\right%
)}F\left({\tfrac{1}{2}a+\tfrac{1}{2},\tfrac{1}{2}a-b+1\atop\tfrac{3}{2}};\left%
(\frac{z+1}{z-1}\right)^{2}\right),\]