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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.29792 |
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| _version_ | 1866913170499567616 |
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| author | Castillo, K. Gordillo-Núñez, G. |
| author_facet | Castillo, K. Gordillo-Núñez, G. |
| contents | The $(-1)$-Jacobi, Bannai-Ito, and $(-1)$-Meixner-Pollaczek polynomials are studied in [Trans. Amer. Math. Soc. 364 (2012), 5491-5507], [Adv. Math. 229 (2012), 2123-2158], and [Stud. Appl. Math. 153 (2024), e12728], respectively, through polynomial eigenfunctions of first-order Dunkl operators. The purpose of the present note is to show that these families are not isolated phenomena, but particular instances of a single alternating mechanism which is most naturally formulated at the level of orthogonality functionals, transpose operators, and structural identities. This functional-analytic point of view also leads to an explicit algorithm which, starting from an ordinary orthogonal polynomial sequence in the quadratic variable, systematises the construction of such $(-1)$-classical families. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_29792 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The manufacture of examples: the $(-1)$-classical orthogonal polynomials Castillo, K. Gordillo-Núñez, G. Classical Analysis and ODEs 42C05, 46A13 The $(-1)$-Jacobi, Bannai-Ito, and $(-1)$-Meixner-Pollaczek polynomials are studied in [Trans. Amer. Math. Soc. 364 (2012), 5491-5507], [Adv. Math. 229 (2012), 2123-2158], and [Stud. Appl. Math. 153 (2024), e12728], respectively, through polynomial eigenfunctions of first-order Dunkl operators. The purpose of the present note is to show that these families are not isolated phenomena, but particular instances of a single alternating mechanism which is most naturally formulated at the level of orthogonality functionals, transpose operators, and structural identities. This functional-analytic point of view also leads to an explicit algorithm which, starting from an ordinary orthogonal polynomial sequence in the quadratic variable, systematises the construction of such $(-1)$-classical families. |
| title | The manufacture of examples: the $(-1)$-classical orthogonal polynomials |
| topic | Classical Analysis and ODEs 42C05, 46A13 |
| url | https://arxiv.org/abs/2605.29792 |