Connection formulas for Askey--Wilson polynomials and related expansions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914335214796800 |
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| author | Cohl, Howard S. Groenevelt, Wolter |
| author_facet | Cohl, Howard S. Groenevelt, Wolter |
| contents | We derive and study expansions of and over the Askey--Wilson polynomials. We study these expansions and examine some limits to the continuous dual $q$-Hahn, Al-Salam--Chihara, continuous big $q$-Hermite and continuous $q$-Hermite polynomials and their $q^{-1}$-analogues. The Poisson kernel for the infinite discrete orthogonality relation for the $q^{-1}$-Al-Salam--Chihara polynomials is derived which in a special case reduces to the Gupta--Masson biorthogonal rational ${}_4ϕ_3$-functions. This Poisson kernel implies new infinite series connection relations for the Askey--Wilson polynomials involving these rational ${}_4ϕ_3$-functions. We also consider various interesting limits. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_15824 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Connection formulas for Askey--Wilson polynomials and related expansions Cohl, Howard S. Groenevelt, Wolter Classical Analysis and ODEs 33D15, 33D50 We derive and study expansions of and over the Askey--Wilson polynomials. We study these expansions and examine some limits to the continuous dual $q$-Hahn, Al-Salam--Chihara, continuous big $q$-Hermite and continuous $q$-Hermite polynomials and their $q^{-1}$-analogues. The Poisson kernel for the infinite discrete orthogonality relation for the $q^{-1}$-Al-Salam--Chihara polynomials is derived which in a special case reduces to the Gupta--Masson biorthogonal rational ${}_4ϕ_3$-functions. This Poisson kernel implies new infinite series connection relations for the Askey--Wilson polynomials involving these rational ${}_4ϕ_3$-functions. We also consider various interesting limits. |
| title | Connection formulas for Askey--Wilson polynomials and related expansions |
| topic | Classical Analysis and ODEs 33D15, 33D50 |
| url | https://arxiv.org/abs/2602.15824 |