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

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 Pn(x) + N i=1 ri,nPn−i(x) =Qn(x)+ M i=1 si,nQn−i(x) 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 φ and ψ, with degrees M and N, respectively, such that φu = ψv. 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.  2005 Elsevier Inc. All rights reserved.