q-DEFORMED RATIONALS AND q-CONTINUED FRACTIONS

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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https://hal.archives-ouvertes.fr/hal-02270545
Contributor : Valentin Ovsienko <>
Submitted on : Monday, August 26, 2019 - 2:05:05 AM
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  • HAL Id : hal-02270545, version 1

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Sophie Morier-Genoud, Valentin Ovsienko. q-DEFORMED RATIONALS AND q-CONTINUED FRACTIONS. 2019. ⟨hal-02270545⟩

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