An extension of the associated rational functions on the unit circle

TL;DR

This paper introduces a new class of orthogonal rational functions on the unit circle, generalizing associated rational functions, with explicit representations and properties derived from self-reciprocal coefficients.

Contribution

It extends the theory of orthogonal rational functions by defining a new class via self-reciprocal coefficients and provides explicit formulas and properties for these functions.

Findings

New class includes associated rational functions as a special case

Explicit representations for arbitrary order associated rational functions

Derived properties and Caratheodory functions for the new class

Abstract

A special class of orthogonal rational functions (ORFs) is presented in this paper. Starting with a sequence of ORFs and the corresponding rational functions of the second kind, we define a new sequence as a linear combination of the previous ones, the coefficients of this linear combination being self-reciprocal rational functions. We show that, under very general conditions on the self-reciprocal coefficients, this new sequence satisfies orthogonality conditions as well as a recurrence relation. Further, we identify the Caratheodory function of the corresponding orthogonality measure in terms of such self-reciprocal coefficients. The new class under study includes the associated rational functions as a particular case. As a consequence of the previous general analysis, we obtain explicit representations for the associated rational functions of arbitrary order, as well as for the related Caratheodory function. Such representations are used to find new properties of the associated rational functions.