Bilateral discrete and continuous orthogonality relations in the $q^{-1}$-symmetric Askey scheme
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916417245282304 |
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| author | Cohl, Howard S. Volkmer, Hans |
| author_facet | Cohl, Howard S. Volkmer, Hans |
| contents | In the $q^{-1}$-symmetric Askey scheme, namely the $q^{-1}$-Askey--Wilson, continuous dual $q^{-1}$-Hahn, $q^{-1}$-Al-Salam--Chihara, continuous big $q^{-1}$-Hermite and continuous $q^{-1}$-Hermite polynomials, we compute bilateral discrete and continuous orthogonality relations. We also derive a $q$-beta integral which comes from the continuous orthogonality relation for the $q^{-1}$-Askey--Wilson polynomials. In the $q\to 1^{-}$ limit, this $q$-beta integral corresponds to a beta integral of Ramanujan-type which we present and provide two proofs for. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_00246 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bilateral discrete and continuous orthogonality relations in the $q^{-1}$-symmetric Askey scheme Cohl, Howard S. Volkmer, Hans Classical Analysis and ODEs 33D45, 05A15, 42C05, 05E05, 33D15 In the $q^{-1}$-symmetric Askey scheme, namely the $q^{-1}$-Askey--Wilson, continuous dual $q^{-1}$-Hahn, $q^{-1}$-Al-Salam--Chihara, continuous big $q^{-1}$-Hermite and continuous $q^{-1}$-Hermite polynomials, we compute bilateral discrete and continuous orthogonality relations. We also derive a $q$-beta integral which comes from the continuous orthogonality relation for the $q^{-1}$-Askey--Wilson polynomials. In the $q\to 1^{-}$ limit, this $q$-beta integral corresponds to a beta integral of Ramanujan-type which we present and provide two proofs for. |
| title | Bilateral discrete and continuous orthogonality relations in the $q^{-1}$-symmetric Askey scheme |
| topic | Classical Analysis and ODEs 33D45, 05A15, 42C05, 05E05, 33D15 |
| url | https://arxiv.org/abs/2410.00246 |