  The quantum integer [n]_q is the polynomial 1 + q + q^2 + ... + q^{n-1}, and
the sequence of polynomials { [n]_q }_{n=1}^{\infty} 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.
