Coefficients of $q$-real numbers: their combinatorial meaning and growth

Abstract

A $q$-deformed real number, or ``$q$-real'', was defined by Morier-Genoud and the second author. When $x\in\mathbb{R}$ such that $x\geq0$, the $q$-analogue $[x]_q$ is a power series with integer coefficients in one formal variable~$q$. In general a $q$-real is a formal Laurent series. The main goal of this paper is to study the coefficients of $q$-reals as functions on~$\mathbb{R}$ and give a combinatorial interpretation of these coefficients. This allows us to prove a conjecture studied by several authors stating that the $q$-deformed golden ratio has the smallest radius of convergence among the radii of the $q$-reals associated with positive real numbers. This is a $q$-analogue of the classical Hurwitz theorem. Our approach is combinatorial. We prove that for every real number $x$ in the interval $(1,2)$ the absolute value of each coefficient of the power series representing the $q$-real $[x]_q$ is dominated by the absolute value of the corresponding coefficient of the $q$-deformed golden ratio. The main notion is a certain collection of ordered rooted trees associated with a $q$-real. We prove that the golden ratio corresponds to a universal class of trees.