An ordinary second-order Fuchsian linear differential equation having precisely three singular points:
here are pairwise distinct complex numbers, ( and ) are the characteristic exponents at the singular point (respectively, and ). A Papperitz equation is uniquely determined by the assignment of the singular points and the characteristic exponents. In solving a Papperitz equation (1), use is made of Riemann's notation:
B. Riemann investigated  the problem of finding all many-valued functions , analytic in the extended complex plane, which have the following properties:
a) the function has precisely three singular points ;
b) any three of its branches are connected by a linear equation
with constant coefficients;
c) the function has the simplest singularities at the points ; namely, in a neighbourhood of the point there are two branches and satisfying
where is holomorphic at ; and analogously for and .
Riemann, under certain additional assumptions on the numbers , showed that all such functions can be expressed in terms of hypergeometric functions and that satisfies a linear second-order differential equation with rational coefficients, but did not write this equation out explicitly (see ). The equation in question, (1), was given by E. Papperitz . It is also called the Riemann -equation, the Riemann equation in Papperitz's form and the Riemann equation, and its solutions are called -functions.
The basic properties of the solutions of a Papperitz equation are as follows.
1) A Papperitz equation is invariant under rational-linear transformations: If maps the points to points , then
2) The transformation
transforms a Papperitz equation into a Papperitz equation with the same singular points, but with different characteristic exponents:
3) The hypergeometric equation
is a special case of a Papperitz equation and it corresponds in Riemann's notation to
4) Each solution of a Papperitz equation can be expressed in terms of the hypergeometric function,
under the assumption that is not a negative integer. If none of the differences , , are integers, then interchanging in (2) the positions of and or of and , four solutions of a Papperitz equation are obtained. In addition a Papperitz equation remains unchanged if the positions of the triples , , are rearranged; all these rearrangements provide 24 special solutions of a Papperitz equation (1), which were first obtained by E.E. Kummer .
|||B. Riemann, "Beiträge zur Theorie der durch Gauss'sche Reihe darstellbare Functionen" , Gesammelte math. Werke , Dover, reprint (1953) pp. 67–85|
|||E. Papperitz, "Ueber verwandte -Functionen" Math. Ann. , 25 (1885) pp. 212–221 MR1510304|
|||E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6 MR1424469 MR0595076 MR0178117 MR1519757 Zbl 0951.30002 Zbl 0108.26903 Zbl 0105.26901 Zbl 53.0180.04 Zbl 47.0190.17 Zbl 45.0433.02 Zbl 33.0390.01|
|||V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) MR0100119|
|||E.E. Kummer, "Ueber die hypergeometrische Reihe " J. Reine Angew. Math. , 15 (1836) pp. 39–83; 127–172|
Papperitz equation. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Papperitz_equation&oldid=24524