Product formulas for basic hypergeometric series by evaluations of Askey--Wilson polynomials
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911602158075904 |
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| author | Cohl, Howard Schlosser, Michael |
| author_facet | Cohl, Howard Schlosser, Michael |
| contents | Ismail and Wilson derived a generating function for Askey--Wilson polynomials which is given by a product of $q$-Gauss (Heine) nonterminating basic hypergeometric functions. We provide a generalization of that generating function which contains an extra parameter. A special case gives a closed form summation formula for a quadruple basic hypergeometric sum. We further present new terminating balanced ${}_4ϕ_3$ summations that give rise to $q$-quadratic special values for Askey--Wilson polynomials. We also similarly present new terminating 2-balanced and 3-balanced ${}_4ϕ_3$ summations. Using the Ismail--Wilson generating function combined with explicit summations for terminating balanced basic hypergeometric $_4ϕ_3$ series, we compute new basic hypergeometric product transformations for nonterminating basic hypergeometric series and provide corresponding integral representations. Further new identities are obtained by applying Cayley--Orr type expansion formulas. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_03571 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Product formulas for basic hypergeometric series by evaluations of Askey--Wilson polynomials Cohl, Howard Schlosser, Michael Classical Analysis and ODEs 33D15, 33D50 Ismail and Wilson derived a generating function for Askey--Wilson polynomials which is given by a product of $q$-Gauss (Heine) nonterminating basic hypergeometric functions. We provide a generalization of that generating function which contains an extra parameter. A special case gives a closed form summation formula for a quadruple basic hypergeometric sum. We further present new terminating balanced ${}_4ϕ_3$ summations that give rise to $q$-quadratic special values for Askey--Wilson polynomials. We also similarly present new terminating 2-balanced and 3-balanced ${}_4ϕ_3$ summations. Using the Ismail--Wilson generating function combined with explicit summations for terminating balanced basic hypergeometric $_4ϕ_3$ series, we compute new basic hypergeometric product transformations for nonterminating basic hypergeometric series and provide corresponding integral representations. Further new identities are obtained by applying Cayley--Orr type expansion formulas. |
| title | Product formulas for basic hypergeometric series by evaluations of Askey--Wilson polynomials |
| topic | Classical Analysis and ODEs 33D15, 33D50 |
| url | https://arxiv.org/abs/2411.03571 |