Orthogonal Polynomials Associated with Complementary Chain Sequences

Abstract

Using the minimal parameter sequence of a given chain sequence, we introduce the concept of complementary chain sequences, which we view as perturbations of chain sequences. Using the relation between these complementary chain sequences and the corresponding Verblunsky coefficients, the para-orthogonal polynomials and the associated Szeg\"o polynomials are analyzed. Two illustrations, one involving Gaussian hypergeometric functions and the other involving Carath\'eodory functions are also provided. A connection between these two illustrations by means of complementary chain sequences is also observed.