= Solution
The <Papperitz symbol> describes a homogeneous second-order <Fuchsian differential equation> on the <Riemann sphere>, with three <regular singular points> and the two <indicial exponents> at each. At a finite point $z_j$, an exponent $r$ corresponds locally to a factor $(z-z_j)^r$; at infinity it corresponds to $z^{-r}$. Repeated or resonant exponents can require logarithms. The six exponents obey the <Fuchs relation> that their sum is one. With three singularities there is no independent <accessory parameter>, so these data determine the normalized equation.
A <Möbius transformation of a Papperitz symbol> moves the singular points without changing their exponents. Multiplication of the dependent variable by a local power shifts the two exponents by that power, using the reciprocal local coordinate at infinity.
Set $w=z/(z-1)$ and $v(w)=F(a,c-b,c;w)$. At $w=0,1,\infty$, the exponent pairs for $v$ are respectively $(0,1-c)$, $(0,b-a)$, and $(a,c-b)$. The <Möbius transformation> sends $z=0,1,\infty$ to $w=0,\infty,1$. Before rescaling, $v(w(z))$ therefore has exponent pairs $(0,1-c)$ at zero, $(a,c-b)$ at one, and $(0,b-a)$ at infinity. Multiplication by $(1-z)^{-a}$ subtracts $a$ at one and adds $a$ at infinity. The resulting pairs are
$$
(0,1-c),\quad(0,c-a-b),\quad(a,b),
$$
exactly those of the original <Gauss hypergeometric equation>. Near zero choose the branch with $(1-z)^{-a}=1$ at zero. The resulting solution is analytic there and has value one. Uniqueness of the normalized analytic solution yields
$$
\boxed{F(a,b,c;z)=(1-z)^{-a}F(a,c-b,c;z/(z-1)).}
$$
This is the <Pfaff transformation>. The normalization presupposes the usual admissible parameters, in particular $c\notin\{0,-1,-2,\ldots\}$. Elsewhere the identity holds by <analytic continuation> with compatible branches, or by parameter limits wherever those are defined.
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