$q$-deformed rationals and $q$-continued fractions
Combinatorics
2020-03-11 v3 Number Theory
Abstract
We introduce a notion of -deformed rational numbers and -deformed continued fractions. A -deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the -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 -rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties, -deformation of the Farey graph, matrix presentations and -continuants are given, as well as a relation to the Jones polynomial of rational knots.
Keywords
Cite
@article{arxiv.1812.00170,
title = {$q$-deformed rationals and $q$-continued fractions},
author = {Sophie Morier-Genoud and Valentin Ovsienko},
journal= {arXiv preprint arXiv:1812.00170},
year = {2020}
}