Non-Symmetric Askey--Wilson Shift Operators
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912592032694272 |
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| author | van Horssen, Max Schlösser, Philip |
| author_facet | van Horssen, Max Schlösser, Philip |
| contents | We classify the shift operators for the symmetric Askey-Wilson polynomials and construct shift operators for the non-symmetric Askey-Wilson polynomials using two decompositions of non-symmetric Askey-Wilson polynomials in terms of symmetric ones. These shift operators are difference-reflection operators, and we discuss the conditions under which they restrict to shift operators for the symmetric Askey-Wilson polynomials. We use them to compute the norms of the non-symmetric Askey-Wilson polynomials and compute their specialisations for $q\to1$. These turn out to be shift operators for the non-symmetric Heckman-Opdam polynomials of type $BC_1$ that have recently been found. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_03169 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-Symmetric Askey--Wilson Shift Operators van Horssen, Max Schlösser, Philip Classical Analysis and ODEs Representation Theory 33D45, 20C08 We classify the shift operators for the symmetric Askey-Wilson polynomials and construct shift operators for the non-symmetric Askey-Wilson polynomials using two decompositions of non-symmetric Askey-Wilson polynomials in terms of symmetric ones. These shift operators are difference-reflection operators, and we discuss the conditions under which they restrict to shift operators for the symmetric Askey-Wilson polynomials. We use them to compute the norms of the non-symmetric Askey-Wilson polynomials and compute their specialisations for $q\to1$. These turn out to be shift operators for the non-symmetric Heckman-Opdam polynomials of type $BC_1$ that have recently been found. |
| title | Non-Symmetric Askey--Wilson Shift Operators |
| topic | Classical Analysis and ODEs Representation Theory 33D45, 20C08 |
| url | https://arxiv.org/abs/2412.03169 |