Generalized cyclotomic polynomials associated with regular systems of divisors and arbitrary sets of positive integers

Abstract

We introduce and study the generalized cyclotomic polynomials $\Phi_{A,S,n}(x)$ associated with a regular system $A$ of divisors and an arbitrary set $S$ of positive integers. We show that all of these polynomials have integer coefficients, they can be expressed as the product of certain classical cyclotomic polynomials $\Phi_d(x)$ with $d\mid n$, and enjoy many other properties which are similar to the classical and unitary cases. We also point out some related Menon-type identities. One of them seems to be new even for the cyclotomic polynomials $\Phi_n(x)$.