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 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.
Highlights
Let q be a formal parameter and recall the classical notion of q-integers: [a]q := 1 − qa 1−q = 1+q+ q2 + · · · + qa−1, where a is a nonnegative integer
A similar notion in the case of rational numbers is missing in the literature
The notion of q-rational we introduce has a certain similarity to that of the classical Gaussian q-binomial coefficient r s q, which is a polynomial depending on r and s that q-deforms the usual binomial coefficient r s
Summary
Let q be a formal parameter and recall the classical notion of q-integers:. + q2 + · · · + qa−1, where a is a nonnegative integer. The notion of q-rational we introduce has a certain similarity to that of the classical Gaussian q-binomial coefficient r s q, which is a polynomial depending on r and s that q-deforms the usual binomial coefficient r s. This similarity manifests in the comparison of the weighted Farey sum rule and the q-version of Pascal’s triangle. 1, was observed in [26] as a simple corollary of definition (1.1) This formula allows us to extend the notion of q-deformed rational number to r s. The subsequent paper [26] explores ‘q-deformed real numbers’ based on the notion of q-rationals developed here
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