hypergeometric function

Let $(a,b,c)$ be a triple of complex numbers with $c$ not belonging to the set of negative integers. For a complex number $w$ and a non negative integer $n$, use Pochhammer symbol ${(w)}_n$ , to denote the expression :

$${(w)}_n=w(w+1)\mathrm{\dots }(w+n-1).$$

The Gauss hypergeometric function, ${}_{2}F_1$, is then defined by the following power series expansion :

$${}_{2}F_1(a,b;c;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(a)}_n{(b)}_n}{{(c)}_{n}n!}{z}^n.$$