= Hypergeometric connection formula at one
= Hypergeometric connection formula
{synonym}
Put $d=c-a-b\notin\mathbb Z$ and choose a branch of $(1-z)^d$. The solution regular at zero is
$$
{}_2F_1(a,b;c;z)=\frac{\Gamma(c)\Gamma(d)}{\Gamma(c-a)\Gamma(c-b)}{}_2F_1(a,b;1-d;1-z)+\frac{\Gamma(c)\Gamma(-d)}{\Gamma(a)\Gamma(b)}(1-z)^d{}_2F_1(c-a,c-b;1+d;1-z).
$$
The two <characteristic exponents at a regular singular point> are $0,d$. The <Euler integral for the hypergeometric function> determines the first coefficient by a finite endpoint limit and the second by scaling the singular endpoint; <analytic continuation> extends the parameter ranges.
Back to article page
