Papperitz equation

An ordinary second-order Fuchsian linear differential equation having precisely three singular points:

(1)

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 [1] 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 [1]). The equation in question, (1), was given by E. Papperitz [2]. 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,

(2)

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 [5].

References

[1]	B. Riemann, "Beiträge zur Theorie der durch Gauss'sche Reihe darstellbare Functionen" , Gesammelte math. Werke , Dover, reprint (1953) pp. 67–85
[2]	E. Papperitz, "Ueber verwandte -Functionen" Math. Ann. , 25 (1885) pp. 212–221
[3]	E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6
[4]	V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian)
[5]	E.E. Kummer, "Ueber die hypergeometrische Reihe " J. Reine Angew. Math. , 15 (1836) pp. 39–83; 127–172