Arithmetic density and congruences of $\ell$-regular bipartitions $II$

TL;DR

This paper extends divisibility and congruence results for $ ext{ell}$-regular bipartitions, providing new infinite families of congruences and multiplicative formulas using Hecke eigenform theory and nilpotency of Hecke operators.

Contribution

It improves previous divisibility results for bipartitions and introduces new infinite families of congruences and formulas leveraging advanced modular form techniques.

Findings

$B_{2^{eta}m}(n)$ and $B_{3^{eta}m}(n)$ are almost always divisible by arbitrary powers of 2 and 3.

Established infinite families of congruences for $B_2(n)$ and $B_4(n)$ using Hecke eigenform theory.

Derived congruences modulo arbitrary powers of 2 for $B_{2^{eta}}(n)$ using nilpotency of Hecke operators.

Abstract

Let $ B_{\ell}(n)$ denote the number of $\ell-$regular bipartitions of $n.$ In 2013, Lin \cite{Lin2013} proved a density result for $B_4(n).$ He showed that for any positive integer $k,$ $B_4(n)$ is almost always divisible by $2^k.$ In this article, we improved his result. We prove that $B_{2^{\alpha}m}(n)$ and $B_{3^{\alpha}m}(n)$ are almost always divisible by arbitrary power of $2$ and $3$ respectively. Further, we obtain an infinities families of congruences and multiplicative formulae for $B_2(n)$ and $B_4(n)$ 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 $B_{2^{\alpha}}(n).$