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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2303.14305 |
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| _version_ | 1866914979227107328 |
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| author | Parisi, Ignacio Bono Pacharoni, Inés |
| author_facet | Parisi, Ignacio Bono Pacharoni, Inés |
| contents | The Matrix Bochner Problem aims to classify which weight matrices have their sequence of orthogonal polynomials as eigenfunctions of a second-order differential operator. Casper and Yakimov, in [4], demonstrated that, under certain hypotheses, all solutions to the Matrix Bochner Problem are noncommutative bispectral Darboux transformations of a direct sum of classical scalar weights. This paper aims to provide the first proof that there are solutions to the Matrix Bochner Problem that do not arise through a noncommutative bispectral Darboux transformation of any direct sum of classical scalar weights. This initial example could contribute to a more comprehensive understanding of the general solution to the Matrix Bochner Problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_14305 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Singular examples of the Matrix Bochner Problem Parisi, Ignacio Bono Pacharoni, Inés Classical Analysis and ODEs Rings and Algebras The Matrix Bochner Problem aims to classify which weight matrices have their sequence of orthogonal polynomials as eigenfunctions of a second-order differential operator. Casper and Yakimov, in [4], demonstrated that, under certain hypotheses, all solutions to the Matrix Bochner Problem are noncommutative bispectral Darboux transformations of a direct sum of classical scalar weights. This paper aims to provide the first proof that there are solutions to the Matrix Bochner Problem that do not arise through a noncommutative bispectral Darboux transformation of any direct sum of classical scalar weights. This initial example could contribute to a more comprehensive understanding of the general solution to the Matrix Bochner Problem. |
| title | Singular examples of the Matrix Bochner Problem |
| topic | Classical Analysis and ODEs Rings and Algebras |
| url | https://arxiv.org/abs/2303.14305 |