Ramanujan-style proof of 1"> \(p_{-3}(11n+7) \equiv 0\ (\mathrm{mod\ }11)\)

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• 作者：Bernard L. S. Lin
• 关键词：Jacobi’s four ; square theorem ; Ramanujan’s identity ; Partition congruence
• 刊名：The Ramanujan Journal
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：42
• 期：1
• 页码：223-231
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Number Theory; Field Theory and Polynomials; Combinatorics; Fourier Analysis; Functions of a Complex Variable;
• 出版者：Springer US
• ISSN：1572-9303
• 卷排序：42

文摘

In this note, we establish two identities of \((q;\,q)_\infty ^8\) by using Jacobi’s four-square theorem and two of Ramanujan’s identities. As an important consequence, we present one Ramanujan-style proof of the congruence \(p_{-3}(11n+7)\equiv 0\ (\mathrm{mod\ }11)\), where \(p_{-3}(n)\) denotes the number of 3-color partitions of n.