Mirror symmetric polynomials orthogonal on the unit circle

2025/10/12 by Alexei Zhedanov, Zhedanov, Alexei
Computer Science · Mathematics · #Matrix Theory and Algorithms #Mathematical functions and polynomials #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2510.10447

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.

Citations

• A CMV connection between orthogonal polynomials on the unit circle and the real line
• Persymmetric Jacobi matrices, isospectral deformations and orthogonal polynomials
• CMV matrices and little and big -1 Jacobi polynomials
• Rational interpolation and mixed inverse spectral problem for finite CMV matrices
• An inverse spectral theory for finite CMV matrices
• Mirror Inversion of Quantum States in Linear Registers
• Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle

Related

• Orthogonal polynomials on the unit circle: New results
• OPUC on One Foot
• Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle
• The CMV bispectrality of the Jacobi polynomials on the unit circle
• Self-inversive polynomial and quasi-orthogonality on the unit circle