= Hypergeometric function
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= Gauss hypergeometric function
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The Gauss hypergeometric function is the normalized analytic solution at zero,
$$
{}_2F_1(a,b;c;z)
=\sum_{n=0}^{\infty}\frac{(a)_n(b)_n}{(c)_n}\frac{z^n}{n!},
$$
initially for $|z|<1$ and $c\notin\{0,-1,-2,\ldots\}$, followed by <analytic continuation> where possible.
The ordinary <hypergeometric function> is ${}_2F_1(a,b;c;z)=\sum_{n\geq0}(a)_n(b)_nz^n/((c)_n n!)$ near zero for admissible $c$. It solves the <Gauss hypergeometric equation> and extends by <analytic continuation>.
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