Quadratic addition rules for quantum integers

Abstract

For every positive integer $n$, the quantum integer $[n]_q$ is the polynomial $[n]_q = 1 + q + q^2 + ... + q^{n-1}.$ A quadratic addition rule for quantum integers consists of sequences of polynomials $\mathcal{R}' = \{r'_n(q)\}_{n=1}^{\infty}$, $\mathcal{S}' = \{s'_n(q)\}_{n=1}^{\infty}$, and $\mathcal{T}' = \{t'_{m,n}(q)\}_{m,n=1}^{\infty}$ such that $[m+n]_q = r'_n(q)[m]_q + s'_m(q)[n]_q + t'_{m,n}(q)[m]_q[n]_q$ for all $m$ and $n.$ This paper gives a complete classification of quadratic addition rules, and also considers sequences of polynomials \polf that satisfy the associated functional equation $f_{m+n}(q)= r'_n(q)f_m(q) + s'_m(q)f_n(q) + t'_{m,n}f_m(q)f_n(q).$