= Solution
First assume $a-b\notin\mathbb Z$ and nonsingular parameters, so the two infinity solutions $w_1$ and $w_2$, obtained by exchanging $a,b$, are independent. Write $F=Aw_1+Bw_2$. Since the power series at $1/z=0$ starts at one, $w_1\sim(-z)^{-a}$ and $w_2\sim(-z)^{-b}$.
In the parameter region $\Re(b-a)>0$, let $z\to-\infty$ in the <Pfaff transformation>. The $w_2$ term vanishes after multiplication by $(-z)^a$, while $z/(z-1)\to1$ and $(-z)/(1-z)\to1$. The supplied value at one of the <Gauss hypergeometric function> therefore gives
$$
A=F(a,c-b;c;1)=\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\Gamma(c-a)}.
$$
Analytic continuation in the parameters extends this coefficient to the nonresonant parameter domain. Exchange $a,b$ to determine $B$. Hence the <hypergeometric connection formula at infinity> is
$$
\boxed{F(a,b;c;z)=\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\Gamma(c-a)}w_1(z)+\frac{\Gamma(c)\Gamma(a-b)}{\Gamma(a)\Gamma(c-b)}w_2(z).}
$$
The branch is $|\arg(-z)|<\pi$. The existence of $F(a,b;c;1)$ alone does not exclude integer $a-b$. At such resonant values the two separate displayed terms can be singular; the correct statement takes their combined parameter limit, which can contain logarithms. This qualification is essential, for example when $a=b$.
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