q-实数的系数：其组合意义与增长性
Coefficients of $q$-real numbers: their combinatorial meaning and growth
AI总结：
本文研究q-实数的系数，通过组合方法证明q-变形黄金比例是正实数对应的q-实数中收敛半径最小的，其对应一类通用树。
AI中文摘要：
q-变形实数，或称“q-实数”，由Morier-Genoud与第二作者定义。当x为满足x≥0的实数时，其q-类比物[x]_q是一个含一个形式变量q的整数系数幂级数；一般而言，q-实数是形式洛朗级数。本文主要目标是将q-实数的系数作为ℝ上的函数进行研究，并给出这些系数的组合解释，借此证明多位学者研究过的一个猜想：在与正实数相关的q-实数的收敛半径中，q-变形黄金比例具有最小收敛半径，这是经典Hurwitz定理的q-类比。本文采用组合方法，证明对于区间(1,2)内的任意实数x，代表q-实数[x]_q的幂级数的每个系数的绝对值，都被q-变形黄金比例对应系数的绝对值所控制。核心概念是与q-实数相关的有序根树的特定集合，本文证明黄金比例对应一类通用树。
英文摘要：
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.
