Quantum integers and cyclotomy

Abstract

A sequence of functions F = {f n (q)} ∞ n=1 satisfies the functional equation for multiplication of quantum integers if f mn (q) = f m (q)f n (q m ) 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.

Semidirect Products and Functional Equations for Quantum Multiplication

“…For every positive integer n, the quantum integer [n] q is the polynomial Equivalently, the sequence of polynomials F = {[n] q } ∞ n=1 satisfies the functional equation (1) f mn (q) = f m (q)f n (q m ).…”

“…It is an open problem to classify the sequences of functions (for example, polynomials, rational functions, or formal power series) that satisfy the functional equation (1). These problems have been studied in [1,3,4], and related problems for addition of quantum integers in [2,5,6].…”

Linear quantum addition rules

“…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.…”

“…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 are essentially quotients of products of quantum integers. More precisely, let F = {f n (q)} ∞ n=1 be a nonzero solution of (1) in Q(q), and let supp(F ) be the set of all integers n with f n (q) = 0.…”

“…Let F = {f n (q)} ∞ n=1 be a sequence of polynomials. Nathanson [5] observed that the multiplication ruleinduces a natural multiplication on the sequence of quantum integers, sincefor all positive integers m and n. He asked what sequences F = {f n (q)} ∞ n=1 of polynomials, rational functions, and formal power series satisfy the multiplicative functional equationfor 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.…”