Classification of exceptional Jacobi polynomials
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909628546154496 |
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| author | Garcia-Ferrero, Maria Angeles Gomez-Ullate, David Milson, Robert |
| author_facet | Garcia-Ferrero, Maria Angeles Gomez-Ullate, David Milson, Robert |
| contents | We provide a full classification scheme for exceptional Jacobi operators and polynomials. The classification contains six degeneracy classes according to whether $α,β$ or $α\pmβ$ assume integer values. Exceptional Jacobi operators are in one-to-one correspondence with spectral diagrams, a combinatorial object that describes the number and asymptotic behaviour at the endpoints of $(-1,1)$ of all quasi-rational eigenfunctions of the operator. With a convenient indexing scheme for spectral diagrams, explicit Wronskian and integral construction formulas are given to build the exceptional operators and polynomials from the information encoded in the spectral diagram. In the fully degenerate class $α,β\in\mathbb N_0$ there exist exceptional Jacobi operators with an arbitrary number of continuous parameters. The classification result is achieved by a careful description of all possible rational Darboux transformations that can be performed on exceptional Jacobi operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02656 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classification of exceptional Jacobi polynomials Garcia-Ferrero, Maria Angeles Gomez-Ullate, David Milson, Robert Classical Analysis and ODEs Mathematical Physics 42C05, 33C45, 34M35 We provide a full classification scheme for exceptional Jacobi operators and polynomials. The classification contains six degeneracy classes according to whether $α,β$ or $α\pmβ$ assume integer values. Exceptional Jacobi operators are in one-to-one correspondence with spectral diagrams, a combinatorial object that describes the number and asymptotic behaviour at the endpoints of $(-1,1)$ of all quasi-rational eigenfunctions of the operator. With a convenient indexing scheme for spectral diagrams, explicit Wronskian and integral construction formulas are given to build the exceptional operators and polynomials from the information encoded in the spectral diagram. In the fully degenerate class $α,β\in\mathbb N_0$ there exist exceptional Jacobi operators with an arbitrary number of continuous parameters. The classification result is achieved by a careful description of all possible rational Darboux transformations that can be performed on exceptional Jacobi operators. |
| title | Classification of exceptional Jacobi polynomials |
| topic | Classical Analysis and ODEs Mathematical Physics 42C05, 33C45, 34M35 |
| url | https://arxiv.org/abs/2409.02656 |