\[F\left(a,b;c;z\right)=\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}{\left(b%
\right)_{s}}}{{\left(c\right)_{s}}s!}z^{s}=1+\frac{ab}{c}z+\frac{a(a+1)b(b+1)}%
{c(c+1)2!}z^{2}+\cdots=\frac{\Gamma\left(c\right)}{\Gamma\left(a\right)\Gamma%
\left(b\right)}\sum_{s=0}^{\infty}\frac{\Gamma\left(a+s\right)\Gamma\left(b+s%
\right)}{\Gamma\left(c+s\right)s!}z^{s},\]