Riemann differential equation

A linear homogeneous ordinary differential equation of the second order in the complex plane with three given regular singular points (cf. Regular singular point) $ a $, $ b $ and $ c $ having characteristic exponents $ \alpha , \alpha ^ \prime $, $ \beta , \beta ^ \prime $, $ \gamma , \gamma ^ \prime $ at these points. The general form of such an equation was first given by E. Papperitz, because of which it is also known as a Papperitz equation. Solutions of a Riemann differential equation are written in the form of the so-called Riemann $ P $- function

$$ w = P \left \{ \begin{array}{llll} a & b & c &{} \\ \alpha &\beta &\gamma & z \\ \alpha ^ \prime &\beta ^ \prime &\gamma ^ \prime &{} \\ \end{array} \right \} . $$

Riemann differential equations belong to the class of Fuchsian equations (cf. Fuchsian equation) with three singular points. A particular case of Riemann differential equations is the hypergeometric equation (the singular points are $ 0, 1, \infty $); therefore, a Riemann differential equation itself is sometimes known as a generalized hypergeometric equation. A Riemann differential equation can be reduced to a Pochhammer equation, and its solution can thus be written in the form of an integral over a special contour in the complex plane.

For references see Papperitz equation.