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Hankel continued fractions and Hankel determinants for $q$-deformed metallic numbers

Guo-Niu Han, Emmanuel Pedon

TL;DR

This work analyzes the Hankel-determinant structure of the $q$-deformed metallic numbers $\Phi_n(q)$, defined by a quadratic $q$-continued fraction, and shows that the first $n+2$ shifted Hankel determinant sequences take values in $\{-1,0,1\}$ with explicit periodic (or antiperiodic) behavior. Using the Hankel continued fraction framework (H-fractions) and an algorithmic NextABC procedure, the authors derive a $(6n-4)$-periodic $H$-fraction for $\Phi_n$ and prove that these determinants satisfy a Gale–Robinson three-term recurrence, with all shifts completely determined by the base sequence. They also establish contiguity relations between shifted sequences and obtain explicit determinant formulas, as well as shifted $H$-fraction expansions and modular periodicity results modulo primes. The results connect $q$-deformations of quadratic irrationals to Catalan/Motzkin-type sequences and discrete integrable systems, offering new insights into the interplay between $q$-analogues, Hankel theory, and combinatorial Catalan/Motzkin structures.

Abstract

Fix $n$ a positive integer. Take the $n$-th metallic number $φ_n=\frac{n+\sqrt{n^2+4}}{2}$ (e.g. $φ_1$ is the golden number) and let $Φ_n(q)$ be its $q$-deformation in the sense of S. Morier-Genoud and V. Ovsienko. This is an algebraic continued fraction which admits an expansion into a Taylor series around $q=0$, with integral coefficients. By using the notion of Hankel continued fraction introduced by the first author in 2016 we determine explicitly the first $n+2$ sequences of shifted Hankel determinants of $Φ_n$ and show that they satisfy the following properties: 1) They are periodic and consist of $-1,0,1$ only. 2) They satisfy a three-term Gale-Robinson recurrence, i.e. they form discrete integrable dynamical systems. 3) They are all completely determined by the first sequence. This article thus validates a conjecture formulated by V. Ovsienko and the second author in a recent paper and establishes new connections between $q$-deformations of real numbers and sequences of Catalan or Motzkin numbers.

Hankel continued fractions and Hankel determinants for $q$-deformed metallic numbers

TL;DR

This work analyzes the Hankel-determinant structure of the -deformed metallic numbers , defined by a quadratic -continued fraction, and shows that the first shifted Hankel determinant sequences take values in with explicit periodic (or antiperiodic) behavior. Using the Hankel continued fraction framework (H-fractions) and an algorithmic NextABC procedure, the authors derive a -periodic -fraction for and prove that these determinants satisfy a Gale–Robinson three-term recurrence, with all shifts completely determined by the base sequence. They also establish contiguity relations between shifted sequences and obtain explicit determinant formulas, as well as shifted -fraction expansions and modular periodicity results modulo primes. The results connect -deformations of quadratic irrationals to Catalan/Motzkin-type sequences and discrete integrable systems, offering new insights into the interplay between -analogues, Hankel theory, and combinatorial Catalan/Motzkin structures.

Abstract

Fix a positive integer. Take the -th metallic number (e.g. is the golden number) and let be its -deformation in the sense of S. Morier-Genoud and V. Ovsienko. This is an algebraic continued fraction which admits an expansion into a Taylor series around , with integral coefficients. By using the notion of Hankel continued fraction introduced by the first author in 2016 we determine explicitly the first sequences of shifted Hankel determinants of and show that they satisfy the following properties: 1) They are periodic and consist of only. 2) They satisfy a three-term Gale-Robinson recurrence, i.e. they form discrete integrable dynamical systems. 3) They are all completely determined by the first sequence. This article thus validates a conjecture formulated by V. Ovsienko and the second author in a recent paper and establishes new connections between -deformations of real numbers and sequences of Catalan or Motzkin numbers.

Paper Structure

This paper contains 17 sections, 17 theorems, 198 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background material
  3. S. Morier-Genoud and V. Ovsienko's $q$-numbers
  4. $H$-fractions and their Hankel determinants
  5. The Hankel continued fraction expansion
  6. An algorithm for $H$-fractions
  7. Proof of Theorem \ref{['MainHF']}
  8. A remark on symmetries
  9. Special values and (anti-)periodicity property
  10. Collecting some data
  11. Proof of Theorem \ref{['MainValPer']} in the case $\ell=0$
  12. The Hankel determinants
  13. The Gale-Robinson relations
  14. The shifted case
  15. The $H$-fraction expansion for $\Phi_n^{(\ell)}$ when $\ell\in\{0,1,\ldots,n+1\}$
  16. ...and 2 more sections

Key Result

Theorem 1

Suppose $n≥2$ and define the polynomial The $H$-fraction expansion of the $q$-deformed metallic number $\Phi_n(q)$ defined in defPhin is $(6n-4)$-periodic with offset $1$ and has the following form: where

Figures (5)

  • Figure 5.1: First $70$ Hankel determinants of $\Phi_5$.
  • Figure 6.1: The set $\mathcal{S}\cap\{0,1,\ldots,2n(n+1)\}=\mathcal{S}_0\cup\mathcal{S}_1\cup\mathcal{S}_2$ when $n=5$.
  • Figure 7.1: First $70$ determinants in Hankel sequence $\Delta^{(1)}(\Phi_5)$.
  • Figure 7.2: First $70$ determinants in Hankel sequence $\Delta^{(2)}(\Phi_5)$.
  • Figure 7.3: First $70$ determinants in Hankel sequence $\Delta^{(3)}(\Phi_5)$.

Theorems & Definitions (40)

  • Theorem 1: $H$-fraction expansion for $\Phi_n$
  • Theorem 2: Values and Periodicity
  • Theorem 3: Gale-Robinson recurrence
  • Theorem 4: Contiguity relations
  • Conjecture 5
  • Theorem 6: Periodicity modulo $p$
  • Example 2.1
  • Example 2.2
  • Remark 3.2
  • Definition 3.3
  • ...and 30 more