Para-orthogonal polynomials on the unit circle generated by Kronecker polynomials

TL;DR

This paper explores para-orthogonal polynomials on the unit circle generated by Kronecker polynomials, providing explicit examples and analyzing properties of Sturmian cyclotomic POPUC based on prime factorization.

Contribution

It introduces a new class of Sturmian Kronecker POPUC, offers explicit examples, and analyzes their properties related to cyclotomic polynomials and prime factorization.

Findings

Explicit examples of Sturmian Kronecker POPUC are provided.

Properties of cyclotomic POPUC depend on prime factorization of M.

New formulas for these polynomials are derived.

Abstract

The Kronecker polynomial $K(z)$ is a finite product of cyclotomic polynomials $C_j(z)$. Any Kronecker polynomial $K(z)$ of degree $N+1$ with simple roots on the unit circle generates a finite set $\Phi_0=1, \Phi_1(z), \dots, \Phi_N(z) $ of polynomials (para) orthogonal on the unit circle (POPUC). This set is determined uniquely by the condition $\Phi_N(z) = (N+1)^{-1} K'(z)$. Such set can be called the set of Sturmian Kronecker POPUC. We present several new explicit examples of such POPUC. In particular, we define and analyze properties of the Sturmian cyclotomic POPUC generated by the cyclotomic polynomials $C_M(z)$. Expressions of these polynomials strongly depend on the decomposition of $M$ into prime factors.