Symbolic approach to 2-orthogonal polynomial solutions of a third order differential equation

Abstract

In a recent work, a generic differential operator on the vectorial space of polynomial functions was presented and applied in the study of differential relations fulfilled by polynomial sequences either orthogonal or 2-orthogonal. Considering a third order differential operator that does not increase the degree of polynomials, we search for polynomial eigenfunctions with the help of symbolic computations, assuming that those polynomials constitute a 2-orthogonal polynomial sequence. Two examples are extensively described.