Hankel continued fractions and Hankel determinants for $q$-deformed metallic numbers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911380986134528 |
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| author | Han, Guo-Niu Pedon, Emmanuel |
| author_facet | Han, Guo-Niu Pedon, Emmanuel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05993 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hankel continued fractions and Hankel determinants for $q$-deformed metallic numbers Han, Guo-Niu Pedon, Emmanuel Number Theory Combinatorics 05A30, 11A55, 11C20 (Primary) 11B37, 37J70 (Secondary) 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. |
| title | Hankel continued fractions and Hankel determinants for $q$-deformed metallic numbers |
| topic | Number Theory Combinatorics 05A30, 11A55, 11C20 (Primary) 11B37, 37J70 (Secondary) |
| url | https://arxiv.org/abs/2502.05993 |