Congruences for two-color partitions with odd smallest part
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908870882885632 |
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| author | Andrews, George E. Bachraoui, Mohamed El |
| author_facet | Andrews, George E. Bachraoui, Mohamed El |
| contents | For a fixed positive integer $k$, let $C(k,n)$ denote the number of two-color partitions of $n$ with odd smallest part and restrictions on even parts, and let $C_k(q)$ be its generating function. We show that $C(1,n)\equiv d(2n-1)\pmod{4}$ and obtain congruences modulo $2$ and $4$ for $C(k,n)$ when $k=2,3$. Using $q$-series methods we derive closed formulas for $C_k(q)$ in terms of eta-quotients and formulate Ramanujan-type congruences for the limiting sequence arising from $\lim_{k\to\infty} C_k(q)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_14190 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Congruences for two-color partitions with odd smallest part Andrews, George E. Bachraoui, Mohamed El Number Theory 11P81, 11P83, 05A17, 11F11 For a fixed positive integer $k$, let $C(k,n)$ denote the number of two-color partitions of $n$ with odd smallest part and restrictions on even parts, and let $C_k(q)$ be its generating function. We show that $C(1,n)\equiv d(2n-1)\pmod{4}$ and obtain congruences modulo $2$ and $4$ for $C(k,n)$ when $k=2,3$. Using $q$-series methods we derive closed formulas for $C_k(q)$ in terms of eta-quotients and formulate Ramanujan-type congruences for the limiting sequence arising from $\lim_{k\to\infty} C_k(q)$. |
| title | Congruences for two-color partitions with odd smallest part |
| topic | Number Theory 11P81, 11P83, 05A17, 11F11 |
| url | https://arxiv.org/abs/2410.14190 |