Some congruence properties of a restricted bipartition function \(c_N(n)\) (Q502570)

Summary: Let \(c_N(n)\) denote 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 \(5 \ell\), for any integer \(\ell \geq 1\), by employing Ramanujan's theta-function identities.