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

DOI10.1090/PROC/15915arXiv2107.11430MaRDI QIDQ6373619

Alexei Zhedanov

Publication date: 23 July 2021

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 $Ph{i}_0=1,Ph{i}_1(z),dots,Ph{i}_N(z)$ of polynomials (para) orthogonal on the unit circle (POPUC). This set is determined uniquely by the condition $Ph{i}_N(z)=(N+1{)}^{-1}{K}^{\prime }(z)$. 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.

Mathematics Subject Classification ID

Miscellaneous applications of number theory (11Z05) Other special orthogonal polynomials and functions (33C47)

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