  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 $\Phi_n(x)$.
