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 B_2αm(n) and B_3αm(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 B_2α(n).