General congruences modulo 5 and 7 for colour partitions

Abstract

For any positive integers n and r, let \(p_r(n)\) denotes the number of partitions of n where each part has r distinct colours. Many authors studied the partition function \(p_r(n)\) for particular values of r. In this paper, we prove some general congruences modulo 5 and 7 for the colour partition function \(p_r(n)\) by considering some general values of r. To prove the congruences we employ some q-series identities which are also in the spirit of Ramanujan.