In his seminal work on partitions and divisor functions, MacMahon introduced the following two q-series \(A_k(q)\) and \(B_k(q)\) . Andrews and Rose proved that \(A_k(q)\) and \(C_k(q)\) are quasimodular forms of weight \(\le 2k\) . Recently, Ono and Singh derived two interesting identities involving \(A_k(q)\) and \(C_k(q)\) and proved that the generating functions for the 3-colored partition function \(p_3(n)\) and the overpartition function \(\overline{p}(n)\) have infinitely many closed formulas in terms of \(A_k(q)\) and \(C_k(q)\) . Very recently, Jin, Pandey and Singh derived further such closed formulas for reciprocals of other interesting infinite products. In this paper, we prove that some infinite products can be represented in terms of the convolution sums of \(A_k(q)\) and \(C_k(q)\) by using four nice identities due to Kongsirwong and Liu. In particular, we generalize some interesting results proved by Jin, Pandey and Singh.
