Hypergeometric function

The Gauss hypergeometric function is the normalized analytic solution at zero,initially for $|z|<1$ and $c\notin \{0,-1,-2,\dots \}$, followed by analytic continuation where possible.

The ordinary hypergeometric function is ${}_2{F}_1(a,b;c;z)={\sum }_{n\ge 0}(a{)}_n(b{)}_n{z}^n/((c{)}_{n}n!)$ near zero for admissible $c$. It solves the Gauss hypergeometric equation and extends by analytic continuation.

For $\Re c>\Re b>0$ and a fixed branch away from the cut,Expand $(1-zs{)}^{-a}$ for $|z|<1$ and integrate termwise using the beta function to recover the Gauss hypergeometric function series. Analytic continuation extends the identity in $z$.