CMV matrices and little and big -1 Jacobi polynomials

TL;DR

This paper introduces a novel mapping connecting orthogonal polynomials on the unit circle to those on the real axis, revealing how little and big -1 Jacobi polynomials emerge from Jacobi polynomials via CMV matrix theory.

Contribution

It presents a new map linking unit circle and real axis orthogonal polynomials, incorporating a parameter that results in a linear operator pencil, and explains the origin of -1 Jacobi polynomials.

Findings

The map relates CMV matrices to real orthogonal polynomials.

Little and big -1 Jacobi polynomials are derived from Jacobi polynomials.

The map involves an arbitrary parameter leading to a linear operator pencil.

Abstract

We introduce a new map from polynomials orthogonal on the unit circle to polynomials orthogonal on the real axis. This map is closely related with the theory of CMV matrices. It contains an arbitrary parameter which leads to a linear operator pencil. We show that the little and big -1 Jacobi polynomials are naturally obtained under this map from the Jacobi polynomials on the unit circle.