On the solutions of a functional equation arising from multiplication of quantum integers

Abstract

This paper is the first of several papers in which we prove, for the case where the fields of coefficients are of characteristic zero, four open problems posed in the work of Melvyn Nathanson (2003) [1] concerning the solutions of a functional equation arising from multiplication of quantum integers [n] q = q n−1 + q n−2 + · · · + q + 1.In this paper, we prove one of the problems. The next papers, namely [2-4] by Lan Nguyen, contain the solutions to the other 3 problems.

On the support base of a functional equation arising from multiplication of quantum integers

“…We consider the case where the fields of coefficients of Γ are of characteristic zero. It can be seen from [1][2][3][4] that quantum integers serve as an important source of generators for the solutions Γ above. From [4], it is known that there is no sequence of polynomials, satisfying Functional Equation (2) with support base P containing all primes, which cannot be generated by quantum integers.…”

“…It can be seen from [1][2][3][4] that quantum integers serve as an important source of generators for the solutions Γ above. From [4], it is known that there is no sequence of polynomials, satisfying Functional Equation (2) with support base P containing all primes, which cannot be generated by quantum integers. On the other hand, it is known from [6] that there exist sequences of polynomials satisfying Functional Equation (2) with support base P of finite and infinite cardinality, which cannot be generated by quantum integers.…”

“…From [4], it is known that there is no sequence of polynomials, satisfying Functional Equation (2) with support base P containing all primes, which cannot be generated by quantum integers. On the other hand, it is known from [6] that there exist sequences of polynomials satisfying Functional Equation (2) with support base P of finite and infinite cardinality, which cannot be generated by quantum integers. The goal of this paper is to continue the study, begun in [6], of the limitation of quantum integers as generators for solutions of the functional equations above.…”

“…On the other hand, it is known from [6] that there exist sequences of polynomials satisfying Functional Equation (2) with support base P of finite and infinite cardinality, which cannot be generated by quantum integers. The goal of this paper is to continue the study, begun in [6], of the limitation of quantum integers as generators for solutions of the functional equations above. In particular, we determine explicitly a set of criterions on P that is necessary and sufficient for the existence of a sequence of polynomials, satisfying Functional Equation (2) and having support base P , which cannot be generated by quantum integers.…”

“…The goal of this paper is to continue the study, begun in [6], of the limitation of quantum integers as generators for solutions of the functional equations above. In particular, we determine explicitly a set of criterions on P that is necessary and sufficient for the existence of a sequence of polynomials, satisfying Functional Equation (2) and having support base P , which cannot be generated by quantum integers. First, let us give some basic background and main results from [1,2,4,5], which are relevant to this paper, concerning quantum integers and the functional equation arising from multiplication of these integers.…”