
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 n a x + b TR := { x >  , y > P(x) y } n 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 n 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 ChebTerrab. 