An algorithm for solving some hypergeometric ODE classes.

Lemmus Chan

A simple algorithm for solving a given linear ODE, when it can be
mapped into the (Gauss) hypergeometric equation

          x (1 - x) y'' + ((a + b + 1) x - c) y' + a b y = 0

by means of
                              a x  + b
                 TR := { x -> -------- ,   y -> P(x) y }
                              c x  + d

is presented. In above, P(x) is an arbitrary function, {a,b,c,d} are
arbitrary constants and n is a rational number. The algorithm is based
on the observation that, if such a transformation TR exists, then a
simpler transformation of the form

            power_TR := { x -> x ,    y -> P(x) y },

mapping the given ODE into another one with 3 regular singular points,
also exists. At the core there is the fact that the transformation TR
can be decomposed into power_TR and a Mobius transformation, and the
latter preserves the structure of the singularities of the
hypergeometric equation. Although the idea is simple, it allows to
systematically solve - when such a TR exists - varied ODE problems not
solved by the present Maple or Mathematica ODE solvers.

This is joint work with Edgardo Cheb-Terrab.