Hypergeometric Differential Equation

Abstract

The leading example, in the theory of linear ordinary differential equations with regular singular points of one complex variable, is doubtless the hypergeometrie differential equation:

$$ x(1 - x)\frac{{d^2 u}} {{dx^2 }} + \{ \gamma - (\alpha + \beta + 1)x\} \frac{{du}} {{dx}} - \alpha \beta u = 0, $$

which is a normalized form of linear ordinary differential equations with three regular singular points in the Riemann sphere.