Theta functions and transformations of bilateral basic hypergeometric series
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917493134589952 |
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| author | Zhou, Nian Hong |
| author_facet | Zhou, Nian Hong |
| contents | We establish new transformation formulas involving theta functions and certain bilateral basic hypergeometric series. From these formulas, we construct companion $q$-series for a class of $q$-series such that the asymptotic expansion of their quotient admits a simple closed form. This allows us to prove several conjectures of McIntosh on asymptotic transformations of $q$-series. Moreover, our results extend some identities of Ramanujan and McIntosh. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_05094 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Theta functions and transformations of bilateral basic hypergeometric series Zhou, Nian Hong Number Theory Classical Analysis and ODEs Combinatorics Primary 11B65, Secondary 33D15, 11F27, 05A30 We establish new transformation formulas involving theta functions and certain bilateral basic hypergeometric series. From these formulas, we construct companion $q$-series for a class of $q$-series such that the asymptotic expansion of their quotient admits a simple closed form. This allows us to prove several conjectures of McIntosh on asymptotic transformations of $q$-series. Moreover, our results extend some identities of Ramanujan and McIntosh. |
| title | Theta functions and transformations of bilateral basic hypergeometric series |
| topic | Number Theory Classical Analysis and ODEs Combinatorics Primary 11B65, Secondary 33D15, 11F27, 05A30 |
| url | https://arxiv.org/abs/2605.05094 |