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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.04666 |
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| _version_ | 1866908867228598272 |
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| author | Daniels, Taylor Huber, Timothy McLaughlin, James Ye, Dongxi |
| author_facet | Daniels, Taylor Huber, Timothy McLaughlin, James Ye, Dongxi |
| contents | Let $p \equiv 1 \pmod{4}$ be prime, let $m$ and $n$ be integers such that $p=m^2+n^2$, and let $b$ be a positive integer. Let $Q(z,q) = (z,q/z,q;q)_{\infty}(qz^2,q/z^2;q^2)_{\infty}$ denote the product appearing in the quintuple product identity. We derive explicit formulae for the $p$-dissection of $Q(q^{bm},q^p)Q(q^{bn},q^p)$, and determine sign patterns in length-$p$ arithmetic progressions of the Taylor series coefficients of the associated quotient $Q(q^{bm},q^{p})Q(q^{bn},q^p)/(q^p;q^p)_{\infty}^2$. Some combinatorial applications of the $p$-dissection formulae are also given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_04666 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The $p$-Dissection of a Product of Quintuple Products Daniels, Taylor Huber, Timothy McLaughlin, James Ye, Dongxi Number Theory Combinatorics 11B65, 05A19 Let $p \equiv 1 \pmod{4}$ be prime, let $m$ and $n$ be integers such that $p=m^2+n^2$, and let $b$ be a positive integer. Let $Q(z,q) = (z,q/z,q;q)_{\infty}(qz^2,q/z^2;q^2)_{\infty}$ denote the product appearing in the quintuple product identity. We derive explicit formulae for the $p$-dissection of $Q(q^{bm},q^p)Q(q^{bn},q^p)$, and determine sign patterns in length-$p$ arithmetic progressions of the Taylor series coefficients of the associated quotient $Q(q^{bm},q^{p})Q(q^{bn},q^p)/(q^p;q^p)_{\infty}^2$. Some combinatorial applications of the $p$-dissection formulae are also given. |
| title | The $p$-Dissection of a Product of Quintuple Products |
| topic | Number Theory Combinatorics 11B65, 05A19 |
| url | https://arxiv.org/abs/2603.04666 |