Finding all bessel type solutions for linear differential equations with rational function coefficients

Abstract

A linear differential equation with rational function coefficients has a Bessel type solution when it is solvable in terms of Bv(f), Bv+1(f). For second order equations, with rational function coefficients, f must be a rational function or the square root of a rational function. An algorithm was given by Debeerst, van Hoeij, and Koepf, that can compute Bessel type solutions if and only if f is a rational function. In this paper we extend this work to the square root case, resulting in a complete algorithm to find all Bessel type solutions.