-Deformed Rationals and -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.

Onq-Deformed Real Numbers

“…In this section, we try to give a transparent and self-contained exposition of the notion of q-rational introduced in [6]. We outline an analogy with q-binomial coefficients, and give a recurrent way to compute q-rationals from the Farey graph.…”

“…The unimodality conjecture of [6] states that the coefficients of R and S first grow and then decrease monotonically. Let us mention that the most important properties of q-rationals are the total positivity and the combinatorial interpretation of the coefficients of R and S.…”

“…We take a new and experimental route to introduce a certain version of "q-deformed real numbers", extending q-deformations of rationals introduced in [6]. Given a real number x ≥ 0, we will construct a formal power series with integer coefficients associated with x.…”

“…where R and S both depend on r and s (see [6]). In this paper, we represent these rational functions as Taylor series at q = 0.…”

“…The first one uses the continued fractions and the second 2 × 2 matrices. The reader can find several other equivalent definitions in [6].…”