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

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=qn−1+qn−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.