Rademacher-type exact formula and higher order Turán inequalities for cubic overpartitions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918149695209472 |
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| author | Agarwal, Archit Garg, Meghali Maji, Bibekananda |
| author_facet | Agarwal, Archit Garg, Meghali Maji, Bibekananda |
| contents | In 1918, Hardy and Ramanujan made a breakthrough by developing the circle method to deduce an asymptotic formula for the partition function $p(n)$, which was later refined by Rademacher in 1937 to produce an absolutely convergent series representation for $p(n)$. Since then, Rademacher-type exact formulas for various partition functions have been investigated by many mathematicians. The concept of overpartitions was introduced by Lovejoy and Corteel in 2004. Kim, in 2010, studied an overpartition analogue of cubic partitions, termed as cubic overpartitions. The main objective of this paper is to establish a Rademacher-type exact formula for cubic overpartitions and, as an application, to derive an explicit error term that leads to their log-concavity. Furthermore, applying a result of Griffin, Ono, Rolen, and Zagier, we establish higher-order Turán inequalities for cubic overpartitions. In addition, we obtain log-subadditivity and generalized log-concavity properties for cubic overpartitions inspired by the work of Bessenrodt-Ono and DeSalvo-Pak on the ordinary partition function. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_23151 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rademacher-type exact formula and higher order Turán inequalities for cubic overpartitions Agarwal, Archit Garg, Meghali Maji, Bibekananda Number Theory Primary 05A20, 11N37, 11P82, Secondary 11B57, 11F20 In 1918, Hardy and Ramanujan made a breakthrough by developing the circle method to deduce an asymptotic formula for the partition function $p(n)$, which was later refined by Rademacher in 1937 to produce an absolutely convergent series representation for $p(n)$. Since then, Rademacher-type exact formulas for various partition functions have been investigated by many mathematicians. The concept of overpartitions was introduced by Lovejoy and Corteel in 2004. Kim, in 2010, studied an overpartition analogue of cubic partitions, termed as cubic overpartitions. The main objective of this paper is to establish a Rademacher-type exact formula for cubic overpartitions and, as an application, to derive an explicit error term that leads to their log-concavity. Furthermore, applying a result of Griffin, Ono, Rolen, and Zagier, we establish higher-order Turán inequalities for cubic overpartitions. In addition, we obtain log-subadditivity and generalized log-concavity properties for cubic overpartitions inspired by the work of Bessenrodt-Ono and DeSalvo-Pak on the ordinary partition function. |
| title | Rademacher-type exact formula and higher order Turán inequalities for cubic overpartitions |
| topic | Number Theory Primary 05A20, 11N37, 11P82, Secondary 11B57, 11F20 |
| url | https://arxiv.org/abs/2509.23151 |