Overview
This paper focuses on the coefficients of $q$-deformed real numbers, termed "$q$-reals." Defined by Morier-Genoud and the second author, a $q$-analogue $[x]_q$ for $x \in \mathbb{R}, x \geq 0$ is a power series with integer coefficients in a formal variable $q$. In its general form, a $q$-real is a formal Laurent series. The primary objective of this research is to analyze these coefficients as functions over $\mathbb{R}$ and to establish a combinatorial interpretation for them.
A key outcome of this analysis is the validation of a conjecture, previously explored by multiple researchers, asserting that the $q$-deformed golden ratio possesses the minimal radius of convergence among all $q$-reals associated with positive real numbers. This finding is presented as a $q$-analogue of the classical Hurwitz theorem.
Research Context
The concept of $q$-deformed real numbers, or $q$-reals, serves as the foundation for this study. Specifically, for $x \in \mathbb{R}$ with $x \geq 0$, the $q$-analogue $[x]_q$ is defined as a power series. This power series has integer coefficients and is expressed in a single formal variable, $q$. When considered generally, a $q$-real is characterized as a formal Laurent series. Prior work by Morier-Genoud and the second author established this definition. A specific conjecture concerning the $q$-deformed golden ratio's radius of convergence has been a subject of investigation by several authors, forming a central element of the present research context.
Approach
The methodology employed in this study is combinatorial. The research utilizes a specific collection of ordered rooted trees, associated with each $q$-real, as its main analytical tool. This combinatorial framework enables the investigation of coefficient properties and growth.
Findings
- The coefficients of $q$-reals were studied as functions over $\mathbb{R}$.
- A combinatorial interpretation was provided for these coefficients.
- The conjecture regarding the $q$-deformed golden ratio having the smallest radius of convergence among positive $q$-reals was proven. This result is identified as a $q$-analogue of the classical Hurwitz theorem.
- For any 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 golden ratio is shown to correspond to a universal class of trees within the framework of ordered rooted trees associated with a $q$-real.