On the rational function solutions of functional equations arising from multiplication of quantum integers

We prove results concerning rational function solutions of the functional equations arising from multiplication of quantum integers and thus resolve some problems raised by Melvyn Nathanson. First, we show that the rational function solutions contain more structure than the polynomial solutions in an essential way, namely the non-cyclotomic part of the former is no longer necessarily trivial, which allows us to resolve a problem on the associated Grothendieck group K(Upsilon(P)) of the collection of all polynomial solutions Upsilon(P) with fields of coefficients of characteristic zero and support base P. Second, we show that, contrary to the polynomial solution case, there exists at least one (infinitely many) rational function solution, with support base P containing all primes, which are not constructible from quantum integers. Third, we show that even in the case where the non-cyclotomic part is trivial, rational function solutions are different from polynomial solutions in that there still exist infinitely many rational function solutions to these functional equations with support base P containing all primes and which are not constructible from quantum integers.