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A Papperitz symbol records the three regular singular points of a second-order Fuchsian differential equation and the two local exponents at each. Exponents at infinity refer to solutions behaving as $z^{-\gamma }$, that is, to powers of the local coordinate $1/z$. Their sum satisfies the Fuchs relation: the six exponents add to one. The symbol represents the two-dimensional solution space, not one normalized solution by itself.

The specified exponents give the Gauss hypergeometric equationSet $d=c-a-b$ and let $v=F(c-a,c-b;c;z)$. Its equation isFor $u=(1-z{)}^{d}v$, differentiation gives $u^{\prime }=(1-z{)}^d[v^{\prime }-dv/(1-z)]$ and $u^{\prime \prime }=(1-z{)}^d[v^{\prime \prime }-2d{v}^{\prime }/(1-z)+d(d-1)v/(1-z{)}^2]$. Substitution into the first equation leaves $(1-z{)}^d$ times exactly the second equation: its $v^{\prime }$ coefficient is $c-(a+b+1+2d)z$, and its $v$ coefficient simplifies to $-(c-a)(c-b)$. Hence $u$ solves the same equation as $F(a,b;c;z)$.

The symbol gives the same check geometrically: multiplication by $(1-z{)}^d$ shifts the exponents at one from $\{0,-d\}$ to $\{d,0\}$ and those at infinity from $\{c-a,c-b\}$ to $\{b,a\}$; it does not change the exponents at zero. Choose the branch with $(1-0{)}^d=1$. Then $u$ is analytic at zero and $u(0)=1$. For $c\notin \{0,-1,-2,\dots \}$ the analytic recurrence uniquely determines this normalized solution, provingThe identity continues wherever both sides are defined with compatible branches. At exceptional parameters where the stated normalization does not uniquely define an analytic function, a parameter-continuation convention is needed; the generic proof does not silently assume uniqueness there. This is Euler's hypergeometric transformation.

Write the Papperitz symbol by listing pairs of exponents at $0,1,\infty$. For $F(a,c-b;c;w)$ they are $(0,1-c)$, $(0,b-a)$, and $(a,c-b)$. Under $w=z/(z-1)$ the points $0,1,\infty$ in the $z$-plane correspond respectively to $0,\infty ,1$ in the $w$-plane. Thus the pulled-back exponents are $(0,1-c)$ at zero, $(a,c-b)$ at one, and $(0,b-a)$ at infinity.

Multiplication by $(1-z{)}^{-a}$ shifts the exponents at one by $-a$ and those at infinity by $+a$, yieldingThis is the original Gauss hypergeometric equation. Both functions are analytic and normalized to one at zero, so uniqueness of that normalized local solution proves the Pfaff transformationContinue the identity using the branch $|\arg (1-z)|<\pi$. As usual, the parameters must admit the normalized hypergeometric solution; excluded singular parameter values are treated only when their limiting solution exists.

For $F(a,1+a-c;1+a-b;w)$, the exponents at $w=0,1,\infty$ are $(0,b-a)$, $(0,c-a-b)$, and $(a,1+a-c)$. The substitution $w=1/z$ exchanges zero and infinity, giving $(a,1+a-c)$ at $z=0$, $(0,c-a-b)$ at one, and $(0,b-a)$ at infinity. Multiplication by $(-z{)}^{-a}$ shifts the zero exponents by $-a$ and the infinity exponents by $+a$. The resulting Papperitz symbol is again $(0,1-c)$, $(0,c-a-b)$, $(a,b)$. Therefore $w_1$ satisfies the same hypergeometric equation, with the stated branch $|\arg (-z)|<\pi$.

For the second-order Fuchsian equation represented by the Papperitz symbol, the coefficient of $w^{\prime }$ isThis follows from the sums of local exponents at its two finite regular singular points. Abel's Wronskian identity, obtained by differentiating $w_1{w}_2^{\prime }-w_1^{\prime }{w}_2$ and using the differential equation, gives $W^{\prime }=-pW$. Thus, on every branch domain avoiding zero and one,The normalized expression is locally constant. Its analytic continuation around either singularity is unchanged: the power factors cancel the determinant of the local monodromy, equivalently the integral of $-p$ around that singularity. It therefore defines a single-valued holomorphic function on the punctured plane, locally constant near both punctures. Those punctures are removable, so the normalized expression extends to an entire function. This argument also covers resonant exponents and avoids assuming two pure power solutions in that case.

The chain rule gives $z_x=\sin 2x$, $z_x^2=4z(1-z)$, and $z_{xx}=2(1-2z)$. Thereforewhich, after division, is the printed equation. It has only three regular singular points, $0,1,\infty$, and is thus a Papperitz equation. At zero and one the indicial roots are $0,1/2$. At infinity, inserting $w\sim z^{-\rho }$ gives $\rho =\pm n/2$. Its Papperitz symbol isThe exponent sum is one, as required for a three-singularity second-order equation.

Multiplying the equation by $z(1-z)$ puts it in Gauss hypergeometric equation form with $a=n/2$, $b=-n/2$, $c=1/2$. The solution analytic at zero and normalized to one is ${}_2{F}_1(a,b;c;z)$. Composing it with $z={\sin }^{2}x$ gives an even analytic solution of $w_{xx}+n^{2}w=0$ with $w(0)=1$, $w_x(0)=0$. Uniqueness of this ordinary differential equation impliesThis Gauss hypergeometric function identity initially holds near zero, and on $|x|<\pi /2$ with the usual real branch; it extends by matching analytic continuation. It is not an identity on every real $x$ if the left side is always reevaluated on its principal $z$ branch: for $n=1$ that principal value is $\sqrt{1-{\sin }^{2}x}=|\cos x|$. Continuing the branch consistently restores the stated cosine.

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 areexactly 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 yieldsThis is the Pfaff transformation. The normalization presupposes the usual admissible parameters, in particular $c\notin \{0,-1,-2,\dots \}$. Elsewhere the identity holds by analytic continuation with compatible branches, or by parameter limits wherever those are defined.

Choose a branch of $t=1-z$ with $|\arg t|<\pi$ and write $d=c-a-b\notin \mathbb{Z}$. The Gauss hypergeometric equation isChanging to $t$ turns this into the same equation with third parameter $a+b+1-c=1-d$. Its solution regular at $t=0$ is $y_1=F(a,b,1-d;t)$. Substitution of $y=t^{d}v$ gives the Gauss hypergeometric equation for $v$ with parameters $c-a,c-b,1+d$. Thus $y_2=t^{d}F(c-a,c-b,1+d;t)$ is also a solution. Their distinct nonintegrally separated characteristic exponents at a regular singular point $0,d$ imply linear independence near $z=1$, soThis also follows from the exponent transformation in the Papperitz symbol.

Initially assume $\Re c>\Re b>0$, so the Euler integral for the hypergeometric function applies. If $\Re d>0$, taking $z\to 1$ gives an integral evaluated by the beta function and determinesIndeed $y_1\to 1$ and $y_2\to 0$ there. To find $B$, instead work in the nonempty parameter region $\Re d<0$, put $z=1-ϵ$ and scale the integration variable by $1-s=ϵu$. The singular contribution isThe defining integral of the beta function evaluates this integral to $\Gamma (c-b)\Gamma (-d)/\Gamma (a)$. Since ${ϵ}^{-d}{y}_1\to 0$ and ${ϵ}^{-d}{y}_2\to 1$, it determines $B$. Analytic continuation in the parameters then yields the hypergeometric connection formulaThe usual power-series definition requires $c\notin \{0,-1,-2,\dots \}$; exceptional parameters need a separately specified analytic limit. Reciprocal gamma functions make vanishing coefficients at denominator poles meaningful. The branch must remain fixed when continuing in $z$.

The Papperitz symbolrepresents the two-dimensional solution space of a homogeneous linear differential equation for a second-order Fuchsian differential equation on the Riemann sphere. Its only singularities are the three regular singular points $a,b,c$. The two entries beneath each point are its characteristic exponents at a regular singular point: for example, in the nonresonant case a local basis near $a$ has leading behaviours $(z-a{)}^{\alpha }$ and $(z-a{)}^{{\alpha }^{\prime }}$. If one of the points is infinity, exponent $\rho$ conventionally means behaviour $z^{-\rho }$ there.

The six exponents obey the Fuchs relationSubject to this relation, the symbol determines the corresponding Riemann differential equation; resonant differences between exponents may produce Logarithmic Frobenius solutions without changing the exponent data recorded by the symbol.

Put $y(x)=w(z)$ with $z={\sin }^{2}x$. The chain rule givesand henceAfter substitution and division by $4z(1-z)$, the ordinary differential equation becomesComparison with the Gauss hypergeometric equation gives, up to interchanging $A$ and $B$,

At $z=0$ and $z=1$, the coefficient of $w^{\prime }$ has at most a simple pole and that of $w$ has at most a double pole, so both are regular singular points. The regular singular point at infinity criterion gives the same conclusion at infinity. The corresponding Papperitz symbol isIndeed, the exponent pairs are $(0,1-C)=(0,1/2)$ at zero, $(0,C-A-B)=(0,1/2)$ at one, and $(A,B)=(k/2,-k/2)$ at infinity; their sum also satisfies the Fuchs relation.

The Papperitz symboldescribes the two-dimensional solution space of a second-order homogeneous Fuchsian differential equation on the Riemann sphere with exactly three regular singular points $z_1,z_2,z_3$. Near $z_i$, its two independent Frobenius behaviours are locallysubject to logarithmic modifications in resonant cases. The six characteristic exponents obey the Fuchs relation

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 thatandare 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:The constants depend on $a,b,c$ and the branch convention, but not on $z$. This is the hypergeometric connection formula at infinity.

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.

A Papperitz symbol records the three regular singular points of a second-order Fuchsian equation and the two characteristic exponents at each point. Near $z=1$, substituting $w\sim (z-1{)}^{\rho }$ into the leading terms givesso the exponents are $0$ and $1/2$. The same calculation at $z=-1$ gives $0$ and $1/2$. At infinity, $w\sim z^s$ givesso $s=\pm n$. Since an exponent $\rho$ at infinity denotes behavior $z^{-\rho }$, these are recorded as $-n,n$. Thus the equation has P-symbolThe exponent sum is $1$, as required by the Fuchs relation for three singular points.

With compatible branches and initially near $z=0$,It follows by applying a Möbius transformation and a dependent-variable rescaling to the same Papperitz symbol, then matching the normalized analytic solution at zero.

Riemann's differential equation is a second-order ordinary differential equation with three regular singular points, normalized to zero, one and infinity by a Möbius transformation. Its local characteristic exponents are encoded by the Papperitz symbol and constrained by the Fuchs relation.