-DEFORMED RATIONALS AND -CONTINUED FRACTIONS
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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 identity 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.
- UNIVERSITE PARIS DESCARTES France
- Sorbonne Paris Cité France
- University of Paris France
- French National Centre for Scientific Research France
- Institut de Mathématiques de Jussieu France
q-continued fractions, \(q\)-deformed rational numbers, quiver representations, Continued fractions, [MATH] Mathematics [math], 13F60, QA1-939, FOS: Mathematics, Mathematics - Combinatorics, Number Theory (math.NT), [MATH]Mathematics [math], Farey tessellation, Mathematics - Number Theory, Cluster algebras, \(q\)-deformed continued fractions, q-rationals, 11B57, Farey sequences; the sequences \(1^k, 2^k, \dots\), 004, 11A55, 57M27, total positivity, \(q\)-calculus and related topics, Combinatorics (math.CO), 05A30, Factorials, binomial coefficients, combinatorial functions, Mathematics
q-continued fractions, \(q\)-deformed rational numbers, quiver representations, Continued fractions, [MATH] Mathematics [math], 13F60, QA1-939, FOS: Mathematics, Mathematics - Combinatorics, Number Theory (math.NT), [MATH]Mathematics [math], Farey tessellation, Mathematics - Number Theory, Cluster algebras, \(q\)-deformed continued fractions, q-rationals, 11B57, Farey sequences; the sequences \(1^k, 2^k, \dots\), 004, 11A55, 57M27, total positivity, \(q\)-calculus and related topics, Combinatorics (math.CO), 05A30, Factorials, binomial coefficients, combinatorial functions, Mathematics
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