A functional equation arising from multiplication of quantum integers

For the quantum integer n]_q=1+q+q²+?+qⁿ⁻¹ there is a natural polynomial multiplication such that m]_q⊗_qn]_q=mn]_q. This multiplication leads to the functional equation f_m(q)f_n(q^m)=f_mn(q), defined on a given sequence of polynomials. This paper contains various results concerning the construction and classification of polynomial sequences that satisfy the functional equation, as well open problems that arise from the functional equation.