Polynomial solutions of the third-order Fuchsian linear ODE

Abstract

Polynomial solutions of the hypergeometric equation-Jacobi polynomials constitute an infinite set of orthogonal functions and coincide with eigenfunctions of a singular Sturm-Liouville problem with endpoints of the corresponding interval being regular singularities of the equation (Fuchsian second-order equations with three regular singularities). Among others there are two simple ways of generating these polynomials: i) one way is by using three-term recurrence relations and ii) the other way is by using the Rodrigues formula. The question arises whether it is possible to construct polynomial solutions for the third-order Fuchsian equation with four singularities. These solutions are supposed to be bound at three regular singularities. Taken in general, this problem leads to the necessity to solve algebraic equations of an arbitrary order. However, in particular cases explicit expressions with a generalization of the Rodrigues formula exist. Our starting point is a particular Fuchsian third-order equation with four regular singularities.