Meta algebras and biorthogonal rational functions: the $q$-Hahn case
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arXiv
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| Format: | Preprint |
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2024
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| author | Bernard, Pierre-Antoine Bouziane, Abderahmane Pellerin, Samuel Têtu, Simone Tsujimoto, Satoshi Vinet, Luc Zaimi, Meri Zhedanov, Alexei |
| author_facet | Bernard, Pierre-Antoine Bouziane, Abderahmane Pellerin, Samuel Têtu, Simone Tsujimoto, Satoshi Vinet, Luc Zaimi, Meri Zhedanov, Alexei |
| contents | A unified algebraic interpretation of both finite families of orthogonal polynomials and biorthogonal rational functions of $q$-Hahn type is provided. The approach relies on the meta $q$-Hahn algebra and its finite-dimensional bidiagonal representations. The functions of $q$-Hahn type are identified as overlaps (up to global factors) between bases solving ordinary or generalized eigenvalue problems in the representation of the meta $q$-Hahn algebra. Moreover, (bi)orthogonality relations, recurrence relations, difference equations and some contiguity relations satisfied by these functions are recovered algebraically using the actions of the generators of the meta $q$-Hahn algebra on various bases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_14856 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Meta algebras and biorthogonal rational functions: the $q$-Hahn case Bernard, Pierre-Antoine Bouziane, Abderahmane Pellerin, Samuel Têtu, Simone Tsujimoto, Satoshi Vinet, Luc Zaimi, Meri Zhedanov, Alexei Representation Theory 33D80, 33D45 A unified algebraic interpretation of both finite families of orthogonal polynomials and biorthogonal rational functions of $q$-Hahn type is provided. The approach relies on the meta $q$-Hahn algebra and its finite-dimensional bidiagonal representations. The functions of $q$-Hahn type are identified as overlaps (up to global factors) between bases solving ordinary or generalized eigenvalue problems in the representation of the meta $q$-Hahn algebra. Moreover, (bi)orthogonality relations, recurrence relations, difference equations and some contiguity relations satisfied by these functions are recovered algebraically using the actions of the generators of the meta $q$-Hahn algebra on various bases. |
| title | Meta algebras and biorthogonal rational functions: the $q$-Hahn case |
| topic | Representation Theory 33D80, 33D45 |
| url | https://arxiv.org/abs/2410.14856 |