Ramanujan type of congruences modulo m for (l, m)-regular bipartitions

TL;DR

This paper establishes new infinite families of congruences modulo various primes for the number of (l,m)-regular bipartitions, expanding the understanding of their arithmetic properties using theta function identities.

Contribution

It introduces novel methods employing theta function identities to prove congruences for (l,m)-regular bipartitions modulo primes 3, 7, 11, 13, and 17.

Findings

Proved infinite families of congruences modulo 7, 13, and 17.

Extended known results for congruences modulo 3, 5, and 11.

Demonstrated the effectiveness of theta function identities in partition congruences.

Abstract

Let $B_{l,m}(n)$ denote the number of $(l,m)$-regular bipartitions of $n$. Recently, many authors proved several infinite families of congruences modulo $3$, $5$ and $11$ for $B_{l,m}(n)$. In this paper, using theta function identities to prove infinite families of congruences modulo $m$ for $(l,m)$-regular bipartitions, where $m\in\{7,3,11,13,17\}$.