On the linear functionals associated to linearly related sequences of orthogonal polynomials

Abstract

An inverse problem is solved, by stating that the regular linear functionals u and v associated to linearly related sequences of monic orthogonal polynomials (Pn)n and (Qn)n, respectively, in the sense for all n=0,1,2,... (where ri,n and si,n are complex numbers satisfying some natural conditions), are connected by a rational modification, i.e., there exist polynomials [phi] and [psi], with degrees M and N, respectively, such that [phi]u=[psi]v. We also make some remarks concerning the corresponding direct problem, stating a characterization theorem in the case N=1 and M=2. As an example, we give a linear relation of the above type involving Jacobi polynomials with distinct parameters.