Quadratic addition rules for quantum integers

For every positive integer n, the quantum integer [n]q is the polynomial [n]q=1+q+q2+⋯+qn-1. A quadratic addition rule for quantum integers consists of sequences of polynomials , , and such that for all m and n. This paper gives a complete classification of quadratic addition rules, and also considers sequences of polynomials that satisfy the associated functional equation .