Formal power series arising from multiplication of quantum integers

For the quantum integer [n]q = 1+q+q + · · ·+q there is a natural polynomial multiplication such that [mn]q = [m]q ⊗q [n]q . This multiplication is described by the functional equation fmn(q) = fm(q)fn(q ), defined on a given sequence F = {fn(q)} ∞ n=1 of polynomials such that fn(0) = 1 for all n. If F = {fn(q)} ∞ n=1 is a solution of the functional equation, then there exists a formal power series F (q) such that the sequence {fn(q)} ∞ n=1 converges to F (q). Quantum mulitplication suggests the functional equation f(q)F (q) = F (q), where f(q) is a fixed polynomial or formal power series with constant term f(0) = 1, and F (q) = 1 + ∑∞ k=1 bkq 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 bk 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. 1 Quantum multiplication Nathanson [1, 2] introduced the functional equation for multiplication of quantum integers as follows. 2000 Mathematics Subject Classification: Primary 30B12, 81R50. Secondary 11B13.