On linearly related orthogonal polynomials and their functionals

Abstract

Let {Pn} be a sequence of polynomials orthogonal with respect a linear functional u and {Qn} a sequence of polynomials defined by Pn(x)+snPn−1(x)=Qn(x)+tnQn−1(x). We find necessary and sufficient conditions in order to {Qn} be a sequence of polynomials orthogonal with respect to a linear functional v. Furthermore we prove that the relation between these linear functionals is (x−ã)u=λ(x−a)v. Even more, if u and v are linked in this way we get that {Pn} and {Qn} satisfy a formula as above.