The Liouville–Neumann expansion at a regular singular point

Abstract

We consider homogeneous second order linear differential equations with a regular singular point at x = 0: x 2 y ″+xf(x)y ′+g(x)y = 0, with f ′(x) and g(x) continuous on a closed interval I containing the point x = 0. We show that the Liouville–Neumann expansion is valid for this kind of differential equations. The expansion is uniformly convergent on I, when f(0) is restricted to a certain interval J. The interval J becomes arbitrary large when we take the initial seed of the Liouville–Neumann expansion close enough to the exact solution. As an illustration, we apply this method to the confluent hypergeometric and the Gauss hypergeometric differential equations. We obtain a succession of polynomials that converges to M(a,b; x) absolutely and uniformly on [ − X,X] with fixed X>0. We also obtain a succession of elementary functions that converge to 2 F 1(a, b, c; x) absolutely and uniformly on [ − X, 1 − ϵ] with fixed X>0 and ϵ>0.