On orthogonal polynomials obtained via polynomial mappings

Let (pn)n be a given monic orthogonal polynomial sequence (OPS) and k a fixed positive integer number such that k ≥ 2. We discuss conditions under which this OPS originates from a polynomial mapping in the following sense: to find another monic OPS (qn)n and two polynomials πk and θm , with degrees k and m (resp.), with 0 ≤ 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 is orthogonal in the positive-definite sense, we consider the corresponding inverse problem, giving explicitly the orthogonality measure for the given OPS (pn)n in terms of the orthogonality measure for the OPS (qn)n . Some applications and examples are presented, recovering several known results in a unified way. c ⃝ 2010 Elsevier Inc. All rights reserved.