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(Pn)n and (Qn)n(Qn)n, respectively, in the sensePn(x)+∑i=1Nri,nPn−i(x)=Qn(x)+∑i=1Msi,nQn−i(x) for all n=0,1,2,…n=0,1,2,… (where ri,nri,n and si,nsi,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=ψvϕu=ψv. We also make some remarks concerning the corresponding direct problem, stating a characterization theorem in the case N=1N=1 and M=2M=2. As an example, we give a linear relation of the above type involving Jacobi polynomials with distinct parameters.