Formal power series arising from multiplication of quantum integers

Abstract

For the quantum integer [n]_q = 1+q+q^2+... + q^{n-1} there is a natural polynomial multiplication such that [mn]_q = [m]_q \otimes_q [n]_q. This multiplication is given by the functional equation f_{mn}(q) = f_m(q) f_n(q^m), defined on a sequence {f_n(q)} of polynomials such that f_n(0)=1 for all n. It is proved that if {f_n(q)} is a solution of this functional equation, then the sequence {f_n(q)} converges to a formal power series F(q). Quantum mulitplication also leads to the functional equation f(q)F(q^m) = F(q), where f(q) is a fixed polynomial or formal power series with constant term f(0)=1, and F(q)=1+\sum_{k=1}^{\infty}b_kq^k is a formal power series. It is proved that this functional equation has a unique solution F(q) for every polynomial or formal power series f(q). If the degree of f(q)is at most m-1, then there is an explicit formula for the coefficients b_k of F(q) in terms of the coefficients of f(q) and the m-adic representation of k. The paper also contains a review of convergence properties of formal power series with coefficients in an arbitrary field or integeral domain.