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arXiv · 1812.00170

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

Sophie Morier-Genoud·Valentin Ovsienko
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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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BibTeXRIS

Sophie Morier-Genoud, Valentin Ovsienko. 2018-12-01. $q$-deformed rationals and $q$-continued fractions. https://doi.org/10.1017/fms.2020.9

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