Arithmetic properties of overpartition triples

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• 作者：Liu Quan Wang
• 关键词：Partitions ; overpartition triples ; congruences ; theta functions
• 刊名：Acta Mathematica Sinica, English Series
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：33
• 期：1
• 页码：37-50
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society
• ISSN：1439-7617
• 卷排序：33

文摘

Let \({\overline p _3}\left( n \right)\) be the number of overpartition triples of n. By elementary series manipulations, we establish some congruences for \({\overline p _3}\left( n \right)\) modulo small powers of 2, such as $${\overline p _3}\left( {16n + 14} \right) \equiv 0\left( {\bmod 32} \right),{\overline p _3}\left( {8n + 7} \right) \equiv 0\left( {\bmod 64} \right)$$. We also find many arithmetic properties for \({\overline p _3}\left( n \right)\) modulo 7, 9 and 11, involving the following infinite families of Ramanujan-type congruences: for any integers α ≥ 1 and n ≥ 0, we $$\bar p_3 \left( {3^{2\alpha + 1} \left( {3n + 2} \right)} \right) \equiv 0\left( {\bmod 9\cdot2^4 } \right),\bar p_3 \left( {4^{2\alpha - 1} \left( {56n + 49} \right)} \right) \equiv 0\left( {\bmod 7} \right),\bar p_3 \left( {7^{2\alpha + 1} \left( {7n + 3} \right)} \right) \equiv \bar p_3 \left( {7^{2\alpha + 1} \left( {7n + 5} \right)} \right) \equiv \bar p_3 \left( {7^{2\alpha + 1} \left( {7n + 6} \right)} \right) \equiv 0\left( {\bmod 7} \right)$$, and for r ∈ {1, 2, 3, 4, 5, 6}, \({\overline p _3}\left( {11 \cdot {7^{4\alpha - 1}}\left( {7n + r} \right)} \right) \equiv 0\left( {\bmod 11} \right)\).KeywordsPartitionsoverpartition triplescongruencestheta functionsMR(2010) Subject Classification05A1711P83References[1]Andrews, G. E., Hirschhorn, M. D., Sellers, J. A.: Arithmetic properties of partitions with even parts distinct. Ramanujan J., 23, 169–181 (2010)MathSciNetCrossRefMATHGoogle Scholar[2]Berndt, B. C.: Number Theory in the Spirit of Ramanujan, Amer. Math. Soc., Providence, 2006CrossRefMATHGoogle Scholar[3]Berndt, B. C.: Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991CrossRefMATHGoogle Scholar[4]Bringmann, K., Lovejoy, J.: Rank and congruences for overpartition pairs. Int. J. Number Theory, 4, 303–322 (2008)MathSciNetCrossRefMATHGoogle Scholar[5]Chen, S. C.: Congruences for overpartition k-tuples. Integers, 10, 477–484 (2010)MathSciNetCrossRefMATHGoogle Scholar[6]Chen, W. Y. C., Lin, B. L. S.: Arithmetic properties of overpartition pairs. Acta Arith., 151, 263–277 (2012)MathSciNetCrossRefMATHGoogle Scholar[7]Chen, W. Y. C., Sun, L. H., Wang, R.-H., Zhang, L.: Ramanujan-type congruences for overpartitions modulo 5. J. Number Theory, 148, 62–72 (2015)MathSciNetCrossRefMATHGoogle Scholar[8]Chen W. Y. C., Xia, E. X. W.: Proof of a conjecture of Hirschhorn and Sellers on overpartitions. Acta Arith., 163, 59–69 (2014)MathSciNetCrossRefMATHGoogle Scholar[9]Cooper, S.: Sums of five, seven and nine squares. Ramanujan J., 6, 469–490 (2002)MathSciNetCrossRefMATHGoogle Scholar[10]Corteel, S., Lovejoy, J.: Overpartitions. Trans. Amer. Math. Soc., 356, 1623–1635 (2004)MathSciNetCrossRefMATHGoogle Scholar[11]Hirschhorn, M. D., Sellers, J. A.: On representations of a number as a sum of three squares. Discrete Math., 199, 85–101 (1999)MathSciNetCrossRefMATHGoogle Scholar[12]Hirschhorn, M. D., Sellers, J. A.: An infinite family of overpartition congruences modulo 12. Integers, 5, Article A20 (2005)MathSciNetMATHGoogle Scholar[13]Hirschhorn, M. D., Sellers, J. A.: Arithmetic relations for overpartitions. J. Combin. Math. Combin. Comput., 53, 65–73 (2005)MathSciNetMATHGoogle Scholar[14]Kim, B.: The overpartition function modulo 128. Integers, 8, Article A38 (2008)MathSciNetMATHGoogle Scholar[15]Kim, B.: A short note on the overpartition function. Discrete Math., 309, 2528–2532 (2009)MathSciNetCrossRefMATHGoogle Scholar[16]Kim, B.: Overpartition pairs modulo powers of 2. Discrete Math., 311, 835–840 (2011)MathSciNetCrossRefMATHGoogle Scholar[17]Lin, B. L. S.: A new proof of a conjecture of Hirschhorn and Sellers on overpartitions. Ramanujan J., 38 (1), 199–209 (2015)MathSciNetCrossRefMATHGoogle Scholar[18]Keister, D. M., Sellers, J. A., Vary, R. G.: Some arithmetic properties of overpartition tuples. Integers, 9, A17, 181–190 (2009)[19]Lovejoy, J.: Overpartition theorems of the Rogers–Ramanujan type. J. London Math. Soc., 69, 562–574 (2004)MathSciNetCrossRefMATHGoogle Scholar[20]Mahlburg, K.: The overpartition function modulo small powers of 2. Discrete. Math., 286(3), 263–267 (2004)MathSciNetCrossRefMATHGoogle Scholar[21]Wang, L.: Another proof of a conjecture of Hirschhorn and Sellers on overpartitions. J. Integer Seq., 17, Article 14.9.8 (2014)MathSciNetMATHGoogle Scholar[22]Xiong, X.: Overpartitions and ternary quadratic forms. Ramanujan J., to appear, doi:10.1007/s11139-016-9773-5Copyright information© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2016Authors and AffiliationsLiu Quan Wang1Email author1.Department of MathematicsNational University of SingaporeSingaporeSingapore About this article CrossMark Publisher Name Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society Print ISSN 1439-8516 Online ISSN 1439-7617 About this journal Reprints and Permissions Article actions .buybox { margin: 16px 0 0; position: relative; } .buybox { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; } .buybox { zoom: 1; } .buybox:after, .buybox:before { content: ''; display: table; } .buybox:after { clear: both; } /*---------------------------------*/ .buybox .buybox__header { border: 1px solid #b3b3b3; border-bottom: 0; padding: 8px 12px; 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