Linear Quantum Addition Rules

The quantum integer [n]q is the polynomial 1+q+q+ · · ·+q. Two sequences of polynomials U = {un(q)}∞n=1 and V = {vn(q)} ∞ n=1 define a linear addition rule ⊕ on a sequence F = {fn(q)}∞n=1 by fm(q) ⊕ fn(q) = un(q)fm(q)+vm(q)fn(q). This is called a quantum addition rule if [m]q⊕[n]q = [m + n]q for all positive integers m and n. In this paper all linear quantum addition rules are determined, and all solutions of the corresponding functional equations fm(q)⊕ fn(q) = fm+n(q) are computed. 1. Multiplication and addition of quantum integers We consider polynomials f(q) with coefficients in a commutative ring with 1. A sequence F = {fn(q)} ∞ n=1 of polynomials is nonzero if fn(q) 6= 0 for some integer n. For every positive integer n, the quantum integer [n]q is the polynomial [n]q = 1 + q + q 2 + · · ·+ q. These polynomials appear in many contexts. In quantum calculus (Cheung-Kac [2]), for example, the q derivative of f(x) = x is f (x) = f(qx) − f(x) qx− x = [n]qx . The quantum integers are ubiquitous in the study of quantum groups (Kassel [3]). Let F = {fn(q)} ∞ n=1 be a sequence of polynomials. Nathanson [5] observed that the multiplication rule fm(q) ∗ fn(q) = fm(q)fn(q ) induces a natural multiplication on the sequence of quantum integers, since [m]q ∗ [n]q = [mn]q for all positive integers m and n. He asked what sequences F = {fn(q)} ∞ n=1 of polynomials, rational functions, and formal power series satisfy the multiplicative functional equation (1) fm(q) ∗ fn(q) = fmn(q) for all positive integers m and n. Borisov, Nathanson, and Wang [1] proved that the only solutions of (1) in the field Q(q) of rational functions with rational coefficients 2000 Mathematics Subject Classification. Primary 11B37, 11P81, 65Q05, 81R50. Secondary 11B13.