A functional equation arising from multiplication of quantum integers

For the quantum integer $[n]_q = 1+q+...+q^{n-1}$ there is a natural polynomial multiplication $*_q$ such that $[m]_q *_q [n]_q = [mn]_q$. This multiplication leads to the functional equation $f_{mn}(q) = f_m(q)f_n(q^m),$ defined on a given sequence $\mathcal(F)=\{f_n(q)\}_{n=1}^{\infty}$ of polynomials. This paper contains various results concerning the classification and construction of polynomial sequences that satisfy the functional equation, as well as a list of open problems that arise fromthe classification.Comment: LaTeX. 18 pages. Revised with minor corrections. To appear in the Journal of Number Theor

Free monoids and forests of rational numbers

The Calkin-Wilf tree is an infinite binary tree whose vertices are the positive rational numbers. Each such number occurs in the tree exactly once and in the form $a/b$, where are $a$ and $b$ are relatively prime positive integers. This tree is associated with the matrices $L_1 = \left( \begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix} \right)$ and $R_1 = \left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right)$, which freely generate the monoid $SL_2(\mathbf{N}_0)$ of $2 \times 2$ matrices with determinant 1 and nonnegative integral coordinates. For other pairs of matrices $L_u$ and $R_v$ that freely generate submonoids of $GL_2(\mathbf{N}_0)$, there are forests of infinitely many rooted infinite binary trees that partition the set of positive rational numbers, and possess a remarkable symmetry property.Comment: 10 page