Arithmetic density and congruences of -regular bipartitions II

Let B_ (n) denote the number of -regular bipartitions of n. In 2013, Lin Lin2013 proved a density result for B₄ (n). He showed that for any positive integer k, B₄ (n) is almost always divisible by 2ᵏ. In this article, we improved his result. We prove that B₂^m (n) and B₃^m (n) are almost always divisible by arbitrary power of 2 and 3 respectively. Further, we obtain an infinities families of congruences and multiplicative formulae for B₂ (n) and B₄ (n) by using Hecke eigenform theory. Next, by using a result of Ono and Taguchi on nilpotency of Hecke operator, we also find an infinite families of congruences modulo arbitrary power of 2 satisfied by B₂^ (n).