Hankel determinants and Jacobi continued fractions for $q$-Euler numbers
The $q$-analogs of Bernoulli and Euler numbers were introduced by Carlitz in 1948. Similar to recent results on the Hankel determinants for the $q$-Bernoulli numbers established by Chapoton and Zeng, we perform a parallel analysis for the $q$-Euler numbers. It is shown that the associated orthogonal...
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Académie des sciences
2024-03-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.569/ |
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| author | Chern, Shane Jiu, Lin |
| author_facet | Chern, Shane Jiu, Lin |
| author_sort | Chern, Shane |
| collection | DOAJ |
| description | The $q$-analogs of Bernoulli and Euler numbers were introduced by Carlitz in 1948. Similar to recent results on the Hankel determinants for the $q$-Bernoulli numbers established by Chapoton and Zeng, we perform a parallel analysis for the $q$-Euler numbers. It is shown that the associated orthogonal polynomials for $q$-Euler numbers are given by a specialization of the big $q$-Jacobi polynomials, thereby leading to their corresponding Jacobi continued fraction expressions, which eventually serve as a key to our determinant evaluations. |
| format | Article |
| id | doaj-art-2516e11505e84bf9b7f727d799d3172a |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-03-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-2516e11505e84bf9b7f727d799d3172a2025-02-07T11:16:26ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-03-01362G220321610.5802/crmath.56910.5802/crmath.569Hankel determinants and Jacobi continued fractions for $q$-Euler numbersChern, Shane0https://orcid.org/0000-0002-8321-1895Jiu, Lin1https://orcid.org/0000-0001-9032-513XDepartment of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, B3H 4R2, CanadaZu Chongzhi Center for Mathematics and Computational Sciences, Duke Kunshan University, Kunshan, Suzhou, Jiangsu Province, 215316, PR ChinaThe $q$-analogs of Bernoulli and Euler numbers were introduced by Carlitz in 1948. Similar to recent results on the Hankel determinants for the $q$-Bernoulli numbers established by Chapoton and Zeng, we perform a parallel analysis for the $q$-Euler numbers. It is shown that the associated orthogonal polynomials for $q$-Euler numbers are given by a specialization of the big $q$-Jacobi polynomials, thereby leading to their corresponding Jacobi continued fraction expressions, which eventually serve as a key to our determinant evaluations.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.569/ |
| spellingShingle | Chern, Shane Jiu, Lin Hankel determinants and Jacobi continued fractions for $q$-Euler numbers Comptes Rendus. Mathématique |
| title | Hankel determinants and Jacobi continued fractions for $q$-Euler numbers |
| title_full | Hankel determinants and Jacobi continued fractions for $q$-Euler numbers |
| title_fullStr | Hankel determinants and Jacobi continued fractions for $q$-Euler numbers |
| title_full_unstemmed | Hankel determinants and Jacobi continued fractions for $q$-Euler numbers |
| title_short | Hankel determinants and Jacobi continued fractions for $q$-Euler numbers |
| title_sort | hankel determinants and jacobi continued fractions for q euler numbers |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.569/ |
| work_keys_str_mv | AT chernshane hankeldeterminantsandjacobicontinuedfractionsforqeulernumbers AT jiulin hankeldeterminantsandjacobicontinuedfractionsforqeulernumbers |