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

TL;DR

This paper introduces a new class of generalized cyclotomic polynomials linked to regular divisor systems and arbitrary positive integer sets, revealing their properties and connections to classical cyclotomic polynomials.

Contribution

It defines and analyzes generalized cyclotomic polynomials associated with regular divisor systems and arbitrary sets, extending classical cyclotomic polynomial theory.

Findings

All polynomials have integer coefficients

They can be expressed as products of classical cyclotomic polynomials

New Menon-type identities related to these polynomials

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 $\Phi_n(x)$.