Semidirect Products and Functional Equations for Quantum Multiplication

Abstract

Abstract. The quantum integer [n]q is the polynomial 1 + q + q 2 + · · · + q n−1 , and the sequence of polynomials {[n]q} ∞ n=1 is a solution of the functional equation fmn(q) = fm(q)fn(q m ). In this paper, semidirect products of semigroups are used to produce families of functional equations that generalize the functional equation for quantum multiplication. Multiplication of quantum integersLet F = {f n (q)} ∞ n=1 be a sequence of functions. We define a binary operation ⊗ on the terms of the sequence F byFor every positive integer n, the quantum integer [n] q is the polynomialEquivalently, the sequence of polynomialssatisfies the functional equationIt 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. Semidirect products of semigroupsLet S and T be semigroups, written multiplicatively, with identity elements e S and e T , respectively, and let Hom(S, T ) denote the set of all semigroup homomorphisms λ : S → T such that λ(e S ) = e T . In this paper, homomorphism always