Bilateral $q$-ultraspherical functions
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912534315925504 |
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| author | Schlosser, Michael J. |
| author_facet | Schlosser, Michael J. |
| contents | We introduce a bilateral extension of the continuous $q$-ultraspherical polynomials which we call bilateral $q$-ultraspherical functions. These functions are given as specific bilateral basic hypergeometric ${}_2ψ_2$ series, they are analytic in a variable $x=\cosθ$ and depend on two parameters $β$ and $γ$ and on a base $q$. For these bilateral $q$-ultraspherical functions we derive a bilateral generating function, find a three-term recurrence relation, explain how they behave under the action of the Askey--Wilson divided difference operator, and show that they satisfy a type of shifted orthogonality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08908 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bilateral $q$-ultraspherical functions Schlosser, Michael J. Classical Analysis and ODEs 33D45 (Primary) 33D15 (Secondary) We introduce a bilateral extension of the continuous $q$-ultraspherical polynomials which we call bilateral $q$-ultraspherical functions. These functions are given as specific bilateral basic hypergeometric ${}_2ψ_2$ series, they are analytic in a variable $x=\cosθ$ and depend on two parameters $β$ and $γ$ and on a base $q$. For these bilateral $q$-ultraspherical functions we derive a bilateral generating function, find a three-term recurrence relation, explain how they behave under the action of the Askey--Wilson divided difference operator, and show that they satisfy a type of shifted orthogonality. |
| title | Bilateral $q$-ultraspherical functions |
| topic | Classical Analysis and ODEs 33D45 (Primary) 33D15 (Secondary) |
| url | https://arxiv.org/abs/2508.08908 |