Quadratic addition rules for quantum integers (Q2490143)

This article is the continuation of the investigation of the second author. It is devoted to the definition of the class of binary operations \(\oplus\) ``quantum addition'' on the sequence \(\{[n]_q\}_{n=1}^\infty\) of quantum integers. The article gives a complete classification of quadratic additional rules, and also considers sequences of polynomials \(F=\{f_n(q)\}_{n+1}^\infty\) that satisfy the associated functional equation \(f_{m+n}(q)=r'_n(q)f_m(q)+s'_m(q)f_n(q)+t'_{m,n} f_m(q) f_n(q).\)