ar X iv : m at h / 03 10 00 5 v 1 [ m at h . N T ] 1 O ct 2 00 3 QUANTUM INTEGERS AND CYCLOTOMY

A sequence of functions F = {fn(q)}∞ n=1 satisfies the functional equation for multiplication of quantum integers if fmn(q) = fm(q)fn(qm) for all positive integers m and n. This paper describes the structure of all sequences of rational functions with coefficients in Q that satisfy this functional equation. 1. The functional equation for multiplication of quantum integers Let N = {1, 2, 3, . . .} denote the positive integers. For every n ∈ N, we define the polynomial [n]q = 1 + q + q 2 + · · ·+ q. This polynomial is called the quantum integer n. The sequence of polynomials {[n]q}n=1 satisfies the following functional equation: (1) fmn(q) = fm(q)fn(q ) for all positive integers m and n. Nathanson [1] asked for a classification of all sequences F = {fn(q)}n=1 of polynomials and of rational functions that satisfy the functional equation (1). The following statements are simple consequences of the functional equation. Proofs can be found in Nathanson [1]. Let F = {fn(q)}n=1 be any sequence of functions that satisfies (1). Then f1(q) = f1(q) 2 = 0 or 1. If f1(q) = 0, then fn(q) = f1(q)fn(q) = 0 for all n ∈ N, and F is a trivial solution of (1). In this paper we consider only nontrivial solutions of the functional equation, that is, sequences F = {fn(q)}n=1 with f1(q) = 1. Let P be a set of prime numbers, and let S(P ) be the multiplicative semigroup of N generated by P . Then S(P ) consists of all integers that can be represented as a product of powers of prime numbers belonging to P . Let F = {fn(q)} ∞ n=1 be a nontrivial solution of (1). We define the support supp(F) = {n ∈ N : fn(q) 6= 0}. There exists a unique set P of prime numbers such that supp(F) = S(P ). Moreover, the sequence F is completely determined by the set {fp(q) : p ∈ P}. Conversely, 2000 Mathematics Subject Classification. Primary 39B05, 81R50, 11R18, 11T22, 11B13.