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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.15735 |
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| _version_ | 1866912283939045376 |
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| author | Santander, John Lopez McLaughlin, Kenneth D. T-R Moll, Victor H. |
| author_facet | Santander, John Lopez McLaughlin, Kenneth D. T-R Moll, Victor H. |
| contents | In this paper we study a family of non-classical Jacobi polynomials with varying parameters of the form $α_n=n+1/2$ and $β_n=-n-1/2$. We obtain global asymptotics for these polynomials, and use this to establish results on the location their zeros. The analysis is based on the Riemann Hilbert formulation of Jacobi polynomials derived from the non-hermitian orthogonality introduced by Kuijlaars, et al. This family of polynomials arise in the symbolic evaluation integrals in the work of Boros and Moll and corresponds to a limitting case, which is not considered in the works of Kuijlaars, et al. A remarkable feature in the analyisis is encountered when performing the local analysis of the RHP near the origin, where the local parametrix introduces a pole. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_15735 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotics and zeros of a special family of Jacobi polynomials Santander, John Lopez McLaughlin, Kenneth D. T-R Moll, Victor H. Classical Analysis and ODEs 41A60 (Primary), 33C45, 34M50 (Secondary) In this paper we study a family of non-classical Jacobi polynomials with varying parameters of the form $α_n=n+1/2$ and $β_n=-n-1/2$. We obtain global asymptotics for these polynomials, and use this to establish results on the location their zeros. The analysis is based on the Riemann Hilbert formulation of Jacobi polynomials derived from the non-hermitian orthogonality introduced by Kuijlaars, et al. This family of polynomials arise in the symbolic evaluation integrals in the work of Boros and Moll and corresponds to a limitting case, which is not considered in the works of Kuijlaars, et al. A remarkable feature in the analyisis is encountered when performing the local analysis of the RHP near the origin, where the local parametrix introduces a pole. |
| title | Asymptotics and zeros of a special family of Jacobi polynomials |
| topic | Classical Analysis and ODEs 41A60 (Primary), 33C45, 34M50 (Secondary) |
| url | https://arxiv.org/abs/2503.15735 |