Semidirect products and functional equations for quantum multiplication

摘要

The quantum integer [n]_q is the polynomial 1 + q + q~2 + ? + q~(n-1), and the sequence of polynomials {[n] _q}_(n = 1)∞ is a solution of the functional equation f_(mn)(q) = f_m(q)f_n(q~m). In this paper, semidirect products of semigroups are used to produce families of functional equations that generalize the functional equation for quantum multiplication.