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

2021/07/23 by Alexei Zhedanov, Zhedanov, Alexei
Chemistry · Materials Science · Mathematics · #11Z05 #33C47 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Liquid Crystal Research Advancements #Mathematical functions and polynomials #Molecular spectroscopy and chirality #math.CA #msc:11Z05 #msc:33C47

paper · pdf · doi:10.48550/arxiv.2107.11430

12 pages, 15 references

arxiv created 2021/07/23 · openalex publication_date 2021/07/23 · arxiv updated 2021/07/27 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

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 Φ₀=1, Φ₁(z), …, Φ_N(z) of polynomials (para) orthogonal on the unit circle (POPUC). This set is determined uniquely by the condition Φ_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.

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