Jacobi三重积的非负系数截断级数
Truncated Series with Nonnegative Coefficients from the Jacobi Triple Product
摘要：
安德鲁斯和梅尔卡研究了欧拉五边形数定理的截断版本，并证明了截断级数的系数是非负的。他们还考虑了由Jacobi三重积恒等式产生的截断级数，并证明其系数是非负的。这个猜想是由郭和曾独立提出的，并由毛和Yee用不同的方法证实。本文给出了与欧拉五边形数定理有关的非负性结果的一个新的组合证明。同时，我们找到了一个类似的结果，截断级数所产生的Jacobi的三重积恒等式以不同的方式。
英文摘要：
Andrews and Merca investigated a truncated version of Euler's pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobi's triple product identity, and they that its coefficients are nonnegative. This conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee using different approaches. In this paper, we provide a new combinatorial proof of their nonnegativity result related to Euler's pentagonal number theorem. Meanwhile, we find an analogous result for a truncated series arising from Jacobi's triple product identity in a different manner.
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