The Habiro ring

Abstract

The Habiro ring is some kind of extension of the ring of polynomials with integer coefficients, in the way that it also contains infinite series with the requirement that they can be evaluated at all roots of unity. Each element of the Habiro ring can be uniquely written as some infinite series in products of the cyclotomic polynomials, called the Habiro expansion. Furthermore, such an element is uniquely determined by its values at (a certain infinite group of) all roots of unity.