On the polynomial and maximal solutions to a functional equation arising from multiplication of quantum integers

We resolve two questions posed by Melvyn Nathanson, Yang Wang, and Alex Borisov concerning solutions with coefficients in Q of the functional equations arising from multiplication of quantum integers. First, we determine the necessary and sufficient criteria for determining when a rational function solution Gamma(P) to these functional equations contains only polynomials. Second, we determine the sets of primes P for which there exist maximal solutions to these functional equations with support bases P. We also give an explicit description of these maximal solutions.