Solutions for Hecke Sum Questions of Banerjee and Bringmann
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866916013398818816 |
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| author | Andrews, George E. Bachraoui, Mohamed El |
| author_facet | Andrews, George E. Bachraoui, Mohamed El |
| contents | The present authors introduced a two-color partition series $S(q)$ and conjectured a Hecke-type formula for the even part of $(q^4;q^4)_\infty S(q)$. Banerjee and Bringmann proved the conjecture by using indefinite theta functions, modular completions, and Sturm's theorem. They also asked whether a direct proof, for instance one based on Bailey-type ideas, could be found, and they suggested that the odd residue classes may be worth studying. We prove a two-variable refinement with an additional parameter $a$.
Our proof relies entirely on $q$-series combined with the Bailey pairs The original even identity and the odd identity then follow as corollaries by letting $a=1$. We also record parameter symmetries and cyclotomic companions, including a vanishing result at $a=i$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_15107 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Solutions for Hecke Sum Questions of Banerjee and Bringmann Andrews, George E. Bachraoui, Mohamed El Number Theory The present authors introduced a two-color partition series $S(q)$ and conjectured a Hecke-type formula for the even part of $(q^4;q^4)_\infty S(q)$. Banerjee and Bringmann proved the conjecture by using indefinite theta functions, modular completions, and Sturm's theorem. They also asked whether a direct proof, for instance one based on Bailey-type ideas, could be found, and they suggested that the odd residue classes may be worth studying. We prove a two-variable refinement with an additional parameter $a$. Our proof relies entirely on $q$-series combined with the Bailey pairs The original even identity and the odd identity then follow as corollaries by letting $a=1$. We also record parameter symmetries and cyclotomic companions, including a vanishing result at $a=i$. |
| title | Solutions for Hecke Sum Questions of Banerjee and Bringmann |
| topic | Number Theory |
| url | https://arxiv.org/abs/2605.15107 |