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$q$-deformed rationals and $q$-continued fractions

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arxiv 1812.00170 v3 pith:WEFUZUDE submitted 2018-12-01 math.CO math.NT

Sophie Morier-Genoud , Valentin Ovsienko This is my paper · ORCID
classification math.COmath.NT
keywords deformedrationalgraphcoefficientscontinuedfareyfractionspascal
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abstract

We introduce a notion of $q$-deformed rational numbers and $q$-deformed continued fractions. A $q$-deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the $q$-deformed Pascal identitiy for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the $q$-rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties, $q$-deformation of the Farey graph, matrix presentations and $q$-continuants are given, as well as a relation to the Jones polynomial of rational knots.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On $q$-deformed real numbers

    math.QA 2019-08 conditional novelty 7.0 of 10

    Every positive real number is assigned a formal power series with integer coefficients, obtained as the stabilized Taylor series of q-deformed rational convergents, and this assignment is shown to be well defined.

  2. Topological model for derived category associated to sphere with four binaries

    math.RT 2026-07 conditional novelty 6.0 of 10

    Indecomposable rigid objects in the derived category of the (2,2,2,2)-weighted projective line are shown to correspond to graded simple arcs on a sphere with four binaries, with Hom-dimensions given by oriented inters...

  3. Nuancing the unicity of $q$-rationals

    math.QA 2026-07 conditional novelty 6.0 of 10

    Exactly two modular-group-equivariant deformations of rationals reproduce the standard q-integers, and the newly identified one has positive coefficients and directly yields Jones polynomials of rational knots.

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