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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/2601.04014 |
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| _version_ | 1866911358895783936 |
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| author | Banerjee, Koustav Bringmann, Kathrin Keith, William J. |
| author_facet | Banerjee, Koustav Bringmann, Kathrin Keith, William J. |
| contents | Recently, Andrews and Bachraoui considered a generating function $F_{k,m}(q)$ associated with certain two-color partitions, and conjectured that this function has non-negative coefficients for $m=1$. They showed this property for $1 \leq k \leq 4$. In this note, we prove that $F_{k,1}(q)$ has non-negative coefficients for $5 \leq k \leq 10$. Moreover, we show that, as $k\to\infty$, $F_{k,1}(q)$ is related to Ramanujan's third order mock theta function $ω(q)$ and to quotients of certain $q$-binomial coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_04014 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On a conjecture of Andrews and Bachraoui Banerjee, Koustav Bringmann, Kathrin Keith, William J. Number Theory 05A17, 11P81 Recently, Andrews and Bachraoui considered a generating function $F_{k,m}(q)$ associated with certain two-color partitions, and conjectured that this function has non-negative coefficients for $m=1$. They showed this property for $1 \leq k \leq 4$. In this note, we prove that $F_{k,1}(q)$ has non-negative coefficients for $5 \leq k \leq 10$. Moreover, we show that, as $k\to\infty$, $F_{k,1}(q)$ is related to Ramanujan's third order mock theta function $ω(q)$ and to quotients of certain $q$-binomial coefficients. |
| title | On a conjecture of Andrews and Bachraoui |
| topic | Number Theory 05A17, 11P81 |
| url | https://arxiv.org/abs/2601.04014 |