The number of tagged parts over the partitions with designated summands

被引：7

作者：

Lin, Bernard L. S. ^{[1
]}

机构：

[1] Jimei Univ, Sch Sci, Xiamen 361021, Peoples R China

来源：

JOURNAL OF NUMBER THEORY | 2018年 / 184卷

基金：

中国国家自然科学基金;

关键词：

Partitions with designated summands; Dissection formula; Congruence; ARITHMETIC PROPERTIES; CONGRUENCES; ANALOGS;

D O I：

10.1016/j.jnt.2017.08.020

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We are concerned with two types of partitions considered by Andrews, Lewis and Lovejoy. One is the partitions with designated summands where exactly one is tagged among parts with equal size. The other is the partitions with designated summands where all parts are odd. In this paper, we study two partition functions PDt(n) and PDOt(n), which count the number of tagged parts over the above two types of partitions respectively. We first give the generating functions of PDt(n) and PDOt(n). Then we establish many congruences modulo small powers of 3 for them. Finally, we pose some problems for future work. (C) 2017 Published by Elsevier Inc.

引用

页码：216 / 234

页数：19

共 20 条

[1] Partitions with designated summands [J].

Andrews, GE ;

Lewis, RP ;

Lovejoy, J .

ACTA ARITHMETICA, 2002, 105 (01) :51-66

[2] MacMahon's sum-of-divisors functions, Chebyshev polynomials, and quasi-modular [J].

Andrews, George E. ;

Rose, Simon C. F. .

JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, 2013, 676 :97-103

[3] Arithmetic properties of partitions with even parts distinct [J].

Andrews, George E. ;

Hirschhorn, Michael D. ;

Sellers, James A. .

RAMANUJAN JOURNAL, 2010, 23 (1-3) :169-181

[4] Analogues of Ramanujan's partition identities and congruences arising from his theta functions and modular equations [J].

Baruah, Nayandeep Deka ;

Ojah, Kanan Kumari .

RAMANUJAN JOURNAL, 2012, 28 (03) :385-407

[5] SOME CONGRUENCES DEDUCIBLE FROM RAMANUJAN'S CUBIC CONTINUED FRACTION [J].

Baruah, Nayandeep Deka ;

Ojah, Kanan Kumari .

INTERNATIONAL JOURNAL OF NUMBER THEORY, 2011, 7 (05) :1331-1343

[6]

Berndt B.C., 1985, Ramanujan's Notebooks

[7]

Berndt B.C., 2006, Number Theory in the Spirit of Ramanujan

[8] On the number of partitions with designated summands [J].

Chen, William Y. C. ;

Ji, Kathy Q. ;

Jin, Hai-Tao ;

Shen, Erin Y. Y. .

JOURNAL OF NUMBER THEORY, 2013, 133 (09) :2929-2938

[9] Formulas for partition k-tuples with t-cores [J].

Chern, Shane .

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 2016, 437 (02) :841-852

[10]

Diamond F., 2005, A First Course in Modular Forms

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