New truncated theorems for three classical theta function identities
New truncated theorems for three classical theta function identities
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DOI：
10.1016/j.ejc.2021.103470
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发表时间：
2022-03
期刊：
影响因子：
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通讯作者：
Ernest X. W. Xia;A. Yee;Xiangui Zhao
中科院分区：
文献类型：
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作者：
Ernest X. W. Xia;A. Yee;Xiangui Zhao
2012年，安德鲁斯和梅尔卡推导出了欧拉五边形数定理的截断版本。他们的工作启发了几位数学家研究截断 theta 级数，其中包括郭和曾，他们研究了高斯的另外两个经典 theta 级数恒等式。在本文中，重新审视欧拉和高斯的这三个theta级数恒等式，我们得到了新的截断定理。作为我们结果的推论，我们获得了涉及配分函数、超配函数和 pod 函数的无限个线性不等式族。这些不等式产生了Andrews-Merca和Guo-Zeng提出的Andrews-Merca和Guo-Zeng关于配分函数的正性结果以及对超配函数的猜想，并由Mao和Yee独立证明。我们还将对我们的结果提供统一的组合处理。
In 2012, Andrews and Merca derived a truncated version of Euler’s pentagonal number theorem. Their work inspired several mathematicians to work on truncated theta series including Guo and Zeng, who examined two other classical theta series identities of Gauss. In this paper, revisiting these three theta series identities of Euler and Gauss, we obtain new truncated theorems. As corollaries of our results, we obtain infinite families of linear inequalities involving the partition function, the overpartition function and the pod function. These inequalities yield the positivity result of Andrews and Merca on the partition function as well as a conjecture on the overpartition function, which was posed by Andrews–Merca and Guo–Zeng, and proved independently by Mao and Yee. We will also provide a unified combinatorial treatment for our results.