On orthogonal polynomials obtained via polynomial mappings

Let (pn)n(pn)n be a given monic orthogonal polynomial sequence (OPS) and kk a fixed positive integer number such that k≥2k≥2. We discuss conditions under which this OPS originates from a polynomial mapping in the following sense: to find another monic OPS (qn)n(qn)n and two polynomials πkπk and θmθm, with degrees kk and mm (resp.), with 0≤m≤k−10≤m≤k−1, such that pnk+m(x)=θm(x)qn(πk(x))(n=0,1,2,…). In this work we establish algebraic conditions for the existence of a polynomial mapping in the above sense. Under such conditions, when (pn)n(pn)n is orthogonal in the positive-definite sense, we consider the corresponding inverse problem, giving explicitly the orthogonality measure for the given OPS (pn)n(pn)n in terms of the orthogonality measure for the OPS (qn)n(qn)n. Some applications and examples are presented, recovering several known results in a unified way.