= Solution
The <Papperitz symbol>
$$
P\left\{
\begin{array}{ccc}
z_1&z_2&z_3\\
\alpha_1&\alpha_2&\alpha_3\\
\beta_1&\beta_2&\beta_3
\end{array}
\right\}
$$
describes the two-dimensional solution space of a second-order homogeneous <Fuchsian differential equation> on the Riemann sphere with exactly three <regular singular point criterion for a second-order equation>[regular singular points] $z_1,z_2,z_3$. Near $z_i$, its two independent Frobenius behaviours are locally
$$
(z-z_i)^{\alpha_i}
\quad\text{and}\quad
(z-z_i)^{\beta_i},
$$
subject to logarithmic modifications in resonant cases. The six <characteristic exponent at a regular singular point>[characteristic exponents] obey the <Fuchs relation>
$$
\sum_{i=1}^3(\alpha_i+\beta_i)=1.
$$
For the <Gauss hypergeometric equation>, the exponents at infinity are $a$ and $b$. Put $w=z^{-1}$. Transforming the Papperitz symbol from $z$ to $w$ shows that
$$
y_a(z)
=z^{-a}F(a,1+a-c;1+a-b;z^{-1})
$$
and
$$
y_b(z)
=z^{-b}F(b,1+b-c;1+b-a;z^{-1})
$$
are local solutions near $z=\infty$, with respective leading behaviours $z^{-a}$ and $z^{-b}$. When $a-b\notin\mathbb Z$, these behaviours are distinct and the two solutions are linearly independent. They therefore form a basis of the solution space on any simply connected common domain with compatible branch choices.
The solution $F(a,b;c;z)$ normalized at zero can be analytically continued into that domain. Since it solves the same second-order equation, it must be a constant linear combination of the basis at infinity:
$$
\boxed{
F(a,b;c;z)
=Az^{-a}F(a,1+a-c;1+a-b;z^{-1})
+Bz^{-b}F(b,1+b-c;1+b-a;z^{-1})
}.
$$
The constants depend on $a,b,c$ and the branch convention, but not on $z$. This is the <hypergeometric connection formula at infinity>.
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