Linear quantum addition rules

Abstract

Abstract. The quantum integer [n]q is the polynomial 1 + q + q 2 + · · · + q n−1 . 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) = f m+n (q) are computed. Multiplication and addition of quantum integersWe consider polynomials f (q) with coefficients in a commutative ring with 1. A sequence F = {f n (q)} ∞ n=1 of polynomials is nonzero if f n (q) = 0 for some integer n. For every positive integer n, the quantum integer [n] q is the polynomialThese polynomials appear in many contexts. In quantum calculus (Cheung-Kac [2]), for example, the q derivative of f (x) = x n isThe quantum integers are ubiquitous in the study of quantum groups (Kassel [3]). 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. Secondary 11B13.

Semidirect Products and Functional Equations for Quantum Multiplication

“…These problems have been studied in [1,3,4], and related problems for addition of quantum integers in [2,5,6]. In this paper we apply the semidirect product of semigroups to produce families of functional equations that generalize the classical functional equation for quantum multiplication.…”

“…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]. In this paper we apply the semidirect product of semigroups to produce families of functional equations that generalize the classical functional equation for quantum multiplication.…”

Quadratic addition rules for quantum integers

“…for all m, n ∈ N. The quantum addition rule (2) is linear. Linear quantum addition rules are considered in [4,5]. …”

On the Classification of Solutions of Quantum Functional Equations with Cyclic and Semi-Cyclic Supports

“…The problems considered in this paper come from the study of symmetry of roots of polynomials called quantum integers and the functional equations arising from their arithmetics in the context of Additive and Combinatorial Number Theory. Such study was initiated by M. B. Nathanson [4,5,6].…”