$q$-DEFORMED RATIONALS AND $q$-CONTINUED FRACTIONS
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 coefficien...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2020-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509420000092/type/journal_article |
| _version_ | 1811156212138377216 |
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| author | SOPHIE MORIER-GENOUD VALENTIN OVSIENKO |
| author_facet | SOPHIE MORIER-GENOUD VALENTIN OVSIENKO |
| author_sort | SOPHIE MORIER-GENOUD |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-10T04:47:45Z |
| format | Article |
| id | doaj.art-f145f72dd3064932a15257b28e2e06ee |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-10T04:47:45Z |
| publishDate | 2020-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-f145f72dd3064932a15257b28e2e06ee2023-03-09T12:34:47ZengCambridge University PressForum of Mathematics, Sigma2050-50942020-01-01810.1017/fms.2020.9$q$-DEFORMED RATIONALS AND $q$-CONTINUED FRACTIONSSOPHIE MORIER-GENOUD0VALENTIN OVSIENKO1https://orcid.org/0000-0003-0146-1573Sorbonne Université, Université Paris Diderot, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, F-75005, Paris, France;Centre national de la recherche scientifique, Laboratoire de Mathématiques, UMR du CNRS 9008, U.F.R. Sciences Exactes et Naturelles Moulin de la Housse - BP 1039 51687 Reims cedex 2, France;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.https://www.cambridge.org/core/product/identifier/S2050509420000092/type/journal_article05A3011A5511B5713F6057M27 |
| spellingShingle | SOPHIE MORIER-GENOUD VALENTIN OVSIENKO $q$-DEFORMED RATIONALS AND $q$-CONTINUED FRACTIONS Forum of Mathematics, Sigma 05A30 11A55 11B57 13F60 57M27 |
| title | $q$-DEFORMED RATIONALS AND $q$-CONTINUED FRACTIONS |
| title_full | $q$-DEFORMED RATIONALS AND $q$-CONTINUED FRACTIONS |
| title_fullStr | $q$-DEFORMED RATIONALS AND $q$-CONTINUED FRACTIONS |
| title_full_unstemmed | $q$-DEFORMED RATIONALS AND $q$-CONTINUED FRACTIONS |
| title_short | $q$-DEFORMED RATIONALS AND $q$-CONTINUED FRACTIONS |
| title_sort | q deformed rationals and q continued fractions |
| topic | 05A30 11A55 11B57 13F60 57M27 |
| url | https://www.cambridge.org/core/product/identifier/S2050509420000092/type/journal_article |
| work_keys_str_mv | AT sophiemoriergenoud qdeformedrationalsandqcontinuedfractions AT valentinovsienko qdeformedrationalsandqcontinuedfractions |