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