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

Abstract

The Kronecker polynomial K ( z ) K(z) is a finite product of cyclotomic polynomials C j ( z ) C_j(z) . Any Kronecker polynomial K ( z ) K(z) of degree N + 1 N+1 with simple roots on the unit circle generates a finite set Φ 0 = 1 \Phi _0=1 , Φ 1 ( z ) \Phi _1(z) , …, Φ N ( z ) \Phi _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 ) \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 ) C_M(z) . Expressions of these polynomials strongly depend on the decomposition of M M into prime factors.