Well-poised basic q-Taylor expansions with complementary remainders and a two-basis kernel
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913163922898944 |
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| author | Abdulsalam, Abdulhafeez A. Schlosser, Michael J. |
| author_facet | Abdulsalam, Abdulhafeez A. Schlosser, Michael J. |
| contents | We prove a nonterminating well-poised basic $q$-Taylor expansion with complementary remainders for a two-basis infinite-product kernel implicitly proposed by the second author in \cite[Sec.~5]{Schlosser2008}. The well-poised parameter $c$ gives the rational $p=0$ basis, while the elliptic nome $p$ is a separate deformation; the infinite expansions treated here are specific to the basic case. We compute the two Taylor coefficient families and show that each one-family Taylor remainder tends to the complementary basis contribution. The proof uses the well-poised Cooper formula, Jackson's terminating ${}_8ϕ_7$ summation, Rogers' ${}_6ϕ_5$ summation, and theta interpolation, but not Bailey's nonterminating ${}_8ϕ_7$ summation, which is recovered as a consequence. We also record two quadratic one-family examples and discuss a multi-kernel outlook. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26011 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Well-poised basic q-Taylor expansions with complementary remainders and a two-basis kernel Abdulsalam, Abdulhafeez A. Schlosser, Michael J. Classical Analysis and ODEs Primary 33D15, Secondary 33D45, 39A13, 41A58 We prove a nonterminating well-poised basic $q$-Taylor expansion with complementary remainders for a two-basis infinite-product kernel implicitly proposed by the second author in \cite[Sec.~5]{Schlosser2008}. The well-poised parameter $c$ gives the rational $p=0$ basis, while the elliptic nome $p$ is a separate deformation; the infinite expansions treated here are specific to the basic case. We compute the two Taylor coefficient families and show that each one-family Taylor remainder tends to the complementary basis contribution. The proof uses the well-poised Cooper formula, Jackson's terminating ${}_8ϕ_7$ summation, Rogers' ${}_6ϕ_5$ summation, and theta interpolation, but not Bailey's nonterminating ${}_8ϕ_7$ summation, which is recovered as a consequence. We also record two quadratic one-family examples and discuss a multi-kernel outlook. |
| title | Well-poised basic q-Taylor expansions with complementary remainders and a two-basis kernel |
| topic | Classical Analysis and ODEs Primary 33D15, Secondary 33D45, 39A13, 41A58 |
| url | https://arxiv.org/abs/2605.26011 |