Quadratic addition rules for three $q$-integers

Abstract

The $q$-integer is the polynomial $[n{]}_q=1+q+q^2+\cdots +q^{n-1}$. For every sequences of polynomials $\mathcal{S}=\{s_m(q){\}}_{m=1}^{\infty }$, $\mathcal{T}=\{t_m(q){\}}_{m=1}^{\infty }$, $\mathcal{U}=\{u_m(q){\}}_{m=1}^{\infty }$ and $\mathcal{V}=\{v_m(q){\}}_{m=1}^{\infty }$, define an addition rule for three $q$-integers by $${\oplus }_{\mathcal{S},\mathcal{T},\mathcal{U},\mathcal{V}}([m{]}_q,[n{]}_q,[k{]}_q)=s_m(q)[m{]}_q+t_m(q)[n{]}_q+u_m(q)[k{]}_q+v_m(q)[n{]}_q[k{]}_q.$$ This is called the first kind of quadratic addition rule for three $q$-integers, if $${\oplus }_{\mathcal{S},\mathcal{T},\mathcal{U},\mathcal{V}}([m{]}_q,[n{]}_q,[k{]}_q)={\left[m+n+k\right]}_q$$ for all positive integers $m$, $n$, $k$. In this paper the first kind of quadratic addition rules for three $q$-integers are determined when $s_m(q)\equiv 1$. Moreover, the solution of the functional equation for a sequence of polynomials $\{f_n(q){\}}_{n=1}^{\infty }$ given by $$f_{m+n+k}(q)=f_m(q)+q^m{f}_n(q)+q^m{f}_k(q)+q^m(q-1)f_n(q)f_k(q)$$ for all positive integers $m$, $n$, $k$, are computed.