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

We introduce and study the generalized cyclotomic polynomials ΦA,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 Φd(x) with d 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 Φₙ(x).