| has gloss | eng: In mathematics, the hypergeometric differential equation is a second-order linear ordinary differential equation (ODE) whose solutions are given by the classical hypergeometric series. Every second-order linear ODE with three regular singular points can be transformed into this equation. The solutions are a special case of a Schwarz–Christoffel mapping to a triangle with circular arcs as edges. These are important because of the role they play in the theory of triangle groups, from which the inverse to Klein's J-invariant may be constructed. Thus, the solutions are coupled to the theory of Fuchsian groups and thus hyperbolic Riemann surfaces. |