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| Content Provider | Springer Nature Link |
|---|---|
| Author | Chen, William Y. C. Fan, Anna R. B. Yu, Rebecca T. |
| Copyright Year | 2014 |
| Abstract | The notion of broken k-diamond partitions was introduced by Andrews and Paule. Let Δ$_{ k }$(n) denote the number of broken k-diamond partitions of n. Andrews and Paule also posed three conjectures on the congruences of Δ$_{2}$(n) modulo 2, 5 and 25. Hirschhorn and Sellers proved the conjectures for modulo 2, and Chan proved the two cases of modulo 5. For the case of modulo 3, Radu and Sellers obtained an infinite family of congruences for Δ$_{2}$(n). In this paper, we obtain two infinite families of congruences for Δ$_{2}$(n) modulo 3 based on a formula of Radu and Sellers, a 3-dissection formula of the generating function of triangular number due to Berndt, and the properties of the U-operator, the V-operator, the Hecke operator and the Hecke eigenform. For example, we find that Δ$_{2}$(243n + 142) ≡ Δ$_{2}$(243n + 223) ≡ 0 (mod 3). The infinite family of Radu and Sellers and the two infinite families derived in this paper have two congruences in common, namely, Δ$_{2}$(27n + 16) ≡ Δ$_{2}$(27n + 25) ≡ 0 (mod 3). |
| Starting Page | 1553 |
| Ending Page | 1560 |
| Page Count | 8 |
| File Format | |
| ISSN | 16747283 |
| Journal | Science in China Series A: Mathematics |
| Volume Number | 57 |
| Issue Number | 8 |
| e-ISSN | 18691862 |
| Language | English |
| Publisher | Science China Press |
| Publisher Date | 2014-06-11 |
| Publisher Place | Heidelberg |
| Access Restriction | One Nation One Subscription (ONOS) |
| Subject Keyword | broken k-diamond partition modular form Ramanujan-type congruence Hecke eigenform Applications of Mathematics |
| Content Type | Text |
| Resource Type | Article |
| Subject | Mathematics |
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