Some Congruences of a Restricted Bipartition Function

TL;DR

This paper investigates congruence properties of a restricted bipartition function, $c_N(n)$, for specific values of N using Ramanujan's theta-function identities, revealing new modular arithmetic patterns.

Contribution

It establishes new congruences for $c_N(n)$ at N=7, 11, and multiples of 5, expanding understanding of bipartition functions with divisibility restrictions.

Findings

Proves congruences for $c_N(n)$ at N=7 and 11.

Derives congruences for $c_N(n)$ when N is a multiple of 5.

Utilizes Ramanujan's theta-function identities to establish these results.

Abstract

Let $c_N(n)$ denotes the number of bipartitions $(\lambda, \mu)$ of a positive integer $n$ subject to the restriction that each part of $\mu$ is divisible by $N$. In this paper, we prove some congruence properties of the function $c_N(n)$ for $N=7$, 11, and $5l$, for any integer $l\ge 1$, by employing Ramanujan's theta-function identities.