Orthogonal Polynomials Associated with Complementary Chain Sequences

TL;DR

This paper introduces complementary chain sequences as perturbations of chain sequences and explores their relation to Verblunsky coefficients, para-orthogonal polynomials, and Szeg"o polynomials, with applications involving hypergeometric and Carathéodory functions.

Contribution

It defines complementary chain sequences and analyzes their connection to orthogonal polynomials and Verblunsky coefficients, providing new insights and illustrations.

Findings

Complementary chain sequences act as perturbations of chain sequences.

Connections established between hypergeometric functions and Carathéodory functions via complementary sequences.

New relations between para-orthogonal and Szeg"o polynomials derived.

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.