Bispectrality of the sieved Jacobi polynomials
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916577502298112 |
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| author | Vinet, Luc Zhedanov, Alexei |
| author_facet | Vinet, Luc Zhedanov, Alexei |
| contents | It is shown that the CMV Laurent polynomials associated to the sieved Jacobi polynomials on the unit circle satisfy an eigenvalue equation with respect to a first order differential operator of Dunkl type. Using this result, the sieved Jacobi polynomials on the real line are found to be eigenfunctions of a Dunkl differential operator of second order. Eigenvalue equations for the sieved ultraspherical polynomials of the first and second kind are obtained as special cases. These results mean that the sieved Jacobi polynomials (either on the unit circle or on the real line) are bispectral. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_12806 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bispectrality of the sieved Jacobi polynomials Vinet, Luc Zhedanov, Alexei Classical Analysis and ODEs Mathematical Physics 33C45, 33C80 It is shown that the CMV Laurent polynomials associated to the sieved Jacobi polynomials on the unit circle satisfy an eigenvalue equation with respect to a first order differential operator of Dunkl type. Using this result, the sieved Jacobi polynomials on the real line are found to be eigenfunctions of a Dunkl differential operator of second order. Eigenvalue equations for the sieved ultraspherical polynomials of the first and second kind are obtained as special cases. These results mean that the sieved Jacobi polynomials (either on the unit circle or on the real line) are bispectral. |
| title | Bispectrality of the sieved Jacobi polynomials |
| topic | Classical Analysis and ODEs Mathematical Physics 33C45, 33C80 |
| url | https://arxiv.org/abs/2501.12806 |