On the rational function solutions of functional equations arising from multiplication of quantum integers

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

“…The results of this paper and those of [9] are also used in [12] to resolve another problem in this area concerning extensions of supports, not necessarily prime subsemigroups of N, of polynomial solutions with fields of coefficients K of characteristic zero. Moreover, we plan to establish in a future paper the analogues of the results in [9] for the rational function solutions with non-prime semigroup supports and fields of coefficients of characteristic zero (see [10,11] for the existence and characterization of rational function solutions with prime semigroup supports and fields of coefficients of characteristic zero).…”

“…,f m (q a i n ) (q) be the root class polynomials associated to α s , f m (q) and α 1 a i n s , f m (q a i n ) , respectively. Then (11) gives: 13) By ( 12) and ( 13), at least one root of f (q) or f n (q a i m ). However, if the former occurs, then it can be deduced that there exists another root class…”

“…Thus ω is an element of α s f m (q) . By (11), ω is a root of f n (q)f m (q a i n ). If ω is a root of f n (q), then it follows from 3) of Proposition 1 that its root class with respect to f n (q) must be β t s f n (q) .…”

“…m ) which consist of exactly all the roots of f β t s ,f n (q) (q a i m ). By (11), ω is a root of f n (q)f m (q a i n ). If ω is a root of f n (q), then ω f n (q) = β t s f n (q) .…”