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Let $\omega$ solve the equation represented by the original symbol and define a dependent-variable rescaling of a Papperitz symbolwith a consistent local choice of branch. Near $a$, the multiplier is a nonzero analytic factor times $(z-a{)}^{\delta }$, so both exponents at $a$ increase by $\delta$. Near $b$, it is a nonzero analytic factor times $(z-b{)}^{-\delta }$, so both exponents at $b$ decrease by $\delta$. It is analytic and nonzero near $c$, so the exponents there are unchanged. Therefore the transformed equation has symbolThe opposite shifts also preserve the Fuchs relation.

A Möbius transformation of a Papperitz symbol changes the independent variable by a Möbius transformation. In particular,maps $a,b,c$ to $0,1,\infty$, respectively. Because a Möbius map is locally biholomorphic away from its pole, it carries each exponent pair with its singular point and givesNow define a dependent-variable rescaling of a Papperitz symbol by writing the old dependent variable asso that $v=u^{-\alpha }(1-u{)}^{-\beta }y$. This dependent-variable rescaling of a Papperitz symbol subtracts $\alpha$ from both exponents at zero, subtracts $\beta$ from both exponents at one, and adds $\alpha +\beta$ to both exponents at infinity. Hence $v$ has symbolDefineThe Fuchs relation gives ${\alpha }^{\prime }-\alpha =1-C$ and ${\beta }^{\prime }-\beta =C-A-B$. After renaming $u$ as $z$, the symbol is thereforewhich is the Gauss hypergeometric equation. Renaming $A,B,C$ as $a,b,c$ gives exactly the displayed symbol in the question.

The Papperitz symbolspecifies a second-order Fuchsian differential equation. The first row lists its three distinct regular singular points $0,1,\infty$; $z$ is the independent variable. The two entries below each singular point are its characteristic exponents. Thus local solutions have leading behaviorsand, with the usual convention at infinity,When an exponent difference is an integer, a logarithmic second solution may replace the naive second Frobenius power. The entries obey the Fuchs relationFor a second-order equation with exactly three regular singular points, these exponent data determine the equation up to multiplication by a nonzero function; there is no accessory parameter. The symbol is therefore the one for the Gauss hypergeometric equation. Its distinguished solution $F(a,b;c;z)$ is the exponent-zero solution analytic at zero and normalized to one there. For the ordinary power-series definition one assumes $c\notin \{0,-1,-2,\dots \}$, with exceptional parameter values handled separately or by continuation.

Now putThe hypergeometric equation in $u$ has exponent pairsThe Möbius transformation of a Papperitz symbol sendsso $Y$ has symbol

On the other hand, $F(a,b;c;z)$ has exponent pairsat $0,1,\infty$. Multiplication by $(1-z{)}^a$ applies the dependent-variable rescaling of a Papperitz symbol: it adds $a$ to both exponents at $z=1$ and subtracts $a$ from both at infinity. Hencehas exactly the same three exponent pairs as $Y$.

Both functions are analytic near $z=0$ and equal one at $z=0$, so uniqueness of the normalized exponent-zero hypergeometric solution gives the Pfaff transformationThe identity first holds near zero with the branch of $(1-z{)}^a$ equal to one there, and then extends by analytic continuation on any domain where compatible branches are chosen.