On radius of convergence of $q$-deformed real numbers

Leclere, Ludivine; Morier-Genoud, Sophie; Ovsienko, Valentin; Веселов, А. П.

doi:10.48550/arxiv.2102.00891

Cited by 5 publications

(9 citation statements)

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“…1 has been proved in [7]. In this paper we solve Conjecture 1.1 for the metallic numbers and its convergence rational sequence.…”

Section: Conjecture 11 ( [7]mentioning

confidence: 92%

“…. , n] q , the case of n = 1, 2 has been examined in [7]. This means that for each q-rational in the q-rational sequence that converges to the q-silver number [2, 2, .…”

Section: Conjecture 11 ( [7]mentioning

confidence: 99%

“…S. Morier-Genoud and V. Ovsienko expand integers to rational numbers and real numbers, and define q-rationals [9] and q-irrationals [10] based on some combinatorial properties of rational numbers and number-theoretic properties of irrational numbers. The q-rationals and the q-irrationals are related to mathematical physics, combinatorics, number theory, quantum algebra, and knot theory, such as calculation recipes of the Jones polynomial and Alexander polynomial of rational knots (see [5], [3] and [11]), quantum Teichmüller spaces [1], and the Markov-Hurwitz approximation theory (see [7] and [2]).…”

Section: Introductionmentioning

confidence: 99%

“…On the study of q-deformation of the real quadratic irrational numbers, L. Leclere and S. Morier-Genoud [6] proved some properties of q-deformation of them corresponding to classical real quadratic irrational numbers. On the other hand, since the q-real numbers are defined as power series in q, under the assumption that q is a complex number, L. Leclere, S. Morier-Genoud, V. Ovsienko, and A. Veselov [7] study its radiuses of convergence and give the following conjecture which can be viewed as a q-deformation of Hurwitz's Irrational Number Theorem.…”

Section: Introductionmentioning

confidence: 99%

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On radiuses of convergence of q-metallic numbers and related q-rational numbers

Ren¹

2021

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The q-rational numbers and the q-irrational numbers are introduced by S. Morier-Genoud and V. Ovsienko. In this paper, we focus on q-real quadratic irrational numbers, especially q-metallic numbers and q-rational sequences which converge to q-metallic numbers, and consider the radiuses of convergence of them when we assume that q is a complex number. We construct two sequences given by recurrence formula as a generalization of the q-deformation of Fibonacci numbers and Pell numbers which are introduced by S. Morier-Genoud and V. Ovsienko. We give an estimation of radiuses of convergence of them, and we solve on conjecture of the lower bound expected which is introduced by L. Leclere, S. Morier-Genoud, V. Ovsienko, and A. Veselov for the metallic numbers and its convergence rational sequence. In addition, we obtain a relationship between the radius of convergence of the [n, n, . . . , n, . . . ] q and [n, n, . . . , n] q in the case of n = 3 and n = 4.

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“…1 has been proved in [7]. In this paper we solve Conjecture 1.1 for the metallic numbers and its convergence rational sequence.…”

Section: Conjecture 11 ( [7]mentioning

confidence: 92%

“…. , n] q , the case of n = 1, 2 has been examined in [7]. This means that for each q-rational in the q-rational sequence that converges to the q-silver number [2, 2, .…”

Section: Conjecture 11 ( [7]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On radiuses of convergence of q-metallic numbers and related q-rational numbers

Ren¹

2021

Preprint

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“…The subject has led to further developments in various directions. Notably there are established links with knots invariants [22], [18], the modular group and the Picard group [20], [33], combinatorics of posets [23], [30], [31], Markov numbers and Markov-Hurwitz approximation theory [9], [17], [19], [21], geometry of Grassmannians [32], triangulated categories [1].…”

Section: Q-analogues Of Rationals and Farey Tessellationmentioning

confidence: 99%

Quantum continuants, quantum rotundus and triangulations of annuli

Leclere¹,

Morier-Genoud²

2022

Preprint

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We give enumerative interpretations of the polynomials arising as numerators and denominators of the q-deformed rational numbers introduced by Morier-Genoud and Ovsienko. The considered polynomials are quantum analogues of the classical continuants and of their cyclically invariant versions called rotundi. The combinatorial models involve triangulations of polygons and annuli. We prove that the quantum continuants are the coarea-generating functions of paths in a triangulated polygon and that the quantum rotundi are the (co)area-generating functions of closed loops on a triangulated annulus.also known as Hirzebruch-Jung continued fractions.

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Quantum Continuants, Quantum Rotundus and Triangulations of Annuli

Leclere,

Morier-Genoud

2023

Electron. J. Combin.

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We give enumerative interpretations of the polynomials arising as numerators and denominators of the $q$-deformed rational numbers introduced by Morier-Genoud and Ovsienko. The considered polynomials are quantum analogues of the classical continuants and of their cyclically invariant versions called rotundi. The combinatorial models involve triangulations of polygons and annuli. We prove that the quantum continuants are the coarea-generating functions of paths in a triangulated polygon and that the quantum rotundi are the (co)area-generating functions of closed loops on a triangulated annulus.

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On radius of convergence of $q$-deformed real numbers

Cited by 5 publications

References 0 publications

On radiuses of convergence of q-metallic numbers and related q-rational numbers

On radiuses of convergence of q-metallic numbers and related q-rational numbers

Quantum continuants, quantum rotundus and triangulations of annuli

Quantum Continuants, Quantum Rotundus and Triangulations of Annuli

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