q-DEFORMED RATIONALS AND q-CONTINUED FRACTIONS

Sophie Morier-Genoud (1) , Valentin Ovsienko (2)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 LMR - Laboratoire de Mathématiques de Reims

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.

Mots clés: quiver representations; Farey tessellation; total positivity; q-continued fractions; q-rationals

Domaines: Mathématiques [math]

Sophie Morier-Genoud, Valentin Ovsienko. q-DEFORMED RATIONALS AND q-CONTINUED FRACTIONS. Forum of Mathematics, Sigma, 2020, 8, pp.e13. ⟨10.1017/fms.2020.9⟩. ⟨hal-02270545⟩