Congruences modulo 4 for broken k-diamond partitions

The notion of broken k-diamond partitions was introduced by Andrews and Paule. Let denote the number of broken k-diamond partitions of n for a fixed positive integer k. Recently, a number of parity results satisfied by for small values of k have been proved by Radu and Sellers and others. However, congruences modulo 4 for are unknown. In this paper, we will prove five congruences modulo 4 for , four infinite families of congruences modulo 4 for and one congruence modulo 4 for by employing theta function identities. Furthermore, we will prove a new parity result for .