Newman cyclotomic polynomials, refinable splines and the Euler binary partition function

Abstract

The class of cyclotomic polynomials (integer polynomials that have primitive complex roots of unity as their roots) is well studied in the literature. We show that its subclass, k-cyclotomic polynomials (k > 2) for which the orders of all complex roots have a common divisor k, possesses some remarkable properties. Such polynomials generate refinable splines, describe the asymptotic growth of the Euler binary partition function, and so on. Moreover, k-cyclotomic polynomials can efficiently be recognized by means of their k-Toeplitz matrices. Special attention is paid to k-cyclotomic Newman (0-1) polynomials, for which we identify one particular family. We prove that all k-cyclotomic polynomials are divisors of polynomials in this family and conjecture that they all actually belong to that family. As an application, we sharpen the asymptotics of the Euler binary partition function and find an explicit formula for it in the case of regular growth.