Ramanujan-type congruences for overpartitions modulo 5

JOURNAL OF NUMBER THEORY

卷:148

Article

Chen, William Y. C.^1,2  Sun, Lisa H.¹  Wang, Rong-Hua¹  Zhang, Li¹ 

[1] Nankai Univ, LPMC TJKLC, Ctr Corabinator, Tianjin 300071, Peoples R China

[2] Tianjin Univ, Ctr Appl Math, Tianjin 300072, Peoples R China

关键词: Overpartition;    Ramanujan-type congruence;    Modular form;    Hecke operator;    Hecke eigenform;   

DOI  :  10.1016/j.jnt.2014.09.017

来源: Elsevier

【 摘 要 】

Let (p) over bar (n) denote the number of overpartitions of n. In this paper, we show that (p) over bar (5n) (-1)(n)(p) over bar (4 . 5n) (mod 5) for n >= 0 and (p) over bar (n) (-1)(n)(p) over bar (4n) (mod 8) for n >= 0 by using the relation of the generating function of (p) over bar (5n) modulo 5 found by Treneer and the 2-adic expansion of the generating function of (p) over bar (n) due to Mahlburg. As a consequence, we deduce that (p) over bar (4(k)(40n + 35)) 0 (mod 40) for n, k >= 0. When k = 0, it was conjectured by Hirschhorn and Sellers, and confirmed by Chen and Xia. Furthermore, applying the Hecke operator on phi(q)(3) and the fact that phi(q)(3) is a Hecke eigenform, we obtain an infinite family of congruences (p) over bar (4(k) . 5l(2)n) 0 (mod 5), where k >= 0 and.e is a prime such that l 3 (mod 5) and (-n/l) = -1. Moreover, we show that (p) over bar (5(2)n) (p) over bar (5(4)n) (mod 5) for n >= 0. So we are led to the congruences (p) over bar (4(k)5(2i+3)(5n +/- 1)) 0 (mod 5) for n, k,i >= 0. In this way, we obtain various Ramanujan-type congruences for (p) over bar (n) modulo 5 such as (p) over bar (45(3n + 1)) 0 (mod 5) and (p) over bar (125(5n +/- 1)) 0 (mod 5) for n >= 0. (C) 2014 Elsevier Inc. All rights reserved.

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