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Mirror symmetric polynomials orthogonal on the unit circle

Alexei Zhedanov

TL;DR

The paper develops mirror-symmetric (persymmetric) orthogonal polynomials on the unit circle (OPUC) by introducing a mirror Verblunsky-dual framework and analyzing associated CMV matrices. It establishes equivalent persymmetry conditions, derives CMV-based algebraic relations analogous to persymmetric Jacobi matrices, and constructs a unitary duality via quasi-reflection operators. Three explicit persymmetric OPUC families are presented, including a new linear-Verblunsky construction tied to symmetric Krawtchouk polynomials, accompanied by spectral and weight formulas. The results include parity-sensitive duality relations for CMV matrices and a discussion of inverse spectral uniqueness, with potential applications to quantum information transfer and further explicit constructions. This work thus extends persymmetric symmetry from OPRL to OPUC, revealing both parallel structure and novel CMV-specific phenomena.

Abstract

We introduce and study a special family of polynomials orthogonal on the unit circle (OPUC). These OPUC satisfy a mirror symmetry property of their Verblunsky coefficients. Several equivalent conditions for the OPUC to be mirror symmetric are presented. Corresponding unitary CMV matrices satisfy simple algebraic relations similar to relations for persymmetric tridiagonal matrices. We present three explicit examples of mirror symmetric OPUC.

Mirror symmetric polynomials orthogonal on the unit circle

TL;DR

The paper develops mirror-symmetric (persymmetric) orthogonal polynomials on the unit circle (OPUC) by introducing a mirror Verblunsky-dual framework and analyzing associated CMV matrices. It establishes equivalent persymmetry conditions, derives CMV-based algebraic relations analogous to persymmetric Jacobi matrices, and constructs a unitary duality via quasi-reflection operators. Three explicit persymmetric OPUC families are presented, including a new linear-Verblunsky construction tied to symmetric Krawtchouk polynomials, accompanied by spectral and weight formulas. The results include parity-sensitive duality relations for CMV matrices and a discussion of inverse spectral uniqueness, with potential applications to quantum information transfer and further explicit constructions. This work thus extends persymmetric symmetry from OPRL to OPUC, revealing both parallel structure and novel CMV-specific phenomena.

Abstract

We introduce and study a special family of polynomials orthogonal on the unit circle (OPUC). These OPUC satisfy a mirror symmetry property of their Verblunsky coefficients. Several equivalent conditions for the OPUC to be mirror symmetric are presented. Corresponding unitary CMV matrices satisfy simple algebraic relations similar to relations for persymmetric tridiagonal matrices. We present three explicit examples of mirror symmetric OPUC.

Paper Structure

This paper contains 8 sections, 15 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Persymmetric Jacobi matrices and polynomials orthogonal on the real line
  3. Polynomials orthogonal on the unit circle
  4. CMV matrices
  5. Mirror dual OPUC and persymmetric OPUC
  6. Explicit examples
  7. Mirror duality relations for CMV matrices
  8. Conclusion

Key Result

Lemma 2.1

The spectral points $x_0, x_1, \dots, x_N$ determine the coefficients $b_0, b_1, \dots, b_N$ and $g_1, g_2, \dots g_N$ of the persymmetric Jacobi matrix $K$ uniquely.

Theorems & Definitions (15)

  • Lemma 2.1
  • Proposition 4.1
  • Proposition 5.1
  • Proposition 5.2
  • Proposition 5.3
  • Proposition 5.4
  • Proposition 6.1
  • Proposition 6.2
  • Proposition 7.1
  • Proposition 7.2
  • ...and 5 more