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 ${\mathcal{R}}^{\prime }=\{r_n^{\prime }(q){\}}_{n=1}^{\infty }$, ${\mathcal{S}}^{\prime }=\{s_n^{\prime }(q){\}}_{n=1}^{\infty }$, and ${\mathcal{T}}^{\prime }=\{t_{m,n}^{\prime }(q){\}}_{m,n=1}^{\infty }$ such that $[m+n{]}_q=r_n^{\prime }(q)[m{]}_q+s_m^{\prime }(q)[n{]}_q+t_{m,n}^{\prime }(q)[m{]}_q[n{]}_q$ 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 $f_{m+n}(q)=r_n^{\prime }(q)f_m(q)+s_m^{\prime }(q)f_n(q)+t_{m,n}^{\prime }{f}_m(q)f_n(q).$

9 pages

References in corpus (2)

• Formal power series arising from multiplication of quantum integers
• Additive number theory and the ring of quantum integers