Two-color partitions are partitions whose parts can be two colors, such as red and green. Let \(\mathcal {L}_d\) denote the set of two-color partitions into numerically distinct parts with the added condition that red parts are at least d larger than the next largest part and green parts are at least \(d+1\) larger than the next largest part, and with no green part \(1_g\) or \((d-1)_g\) . Recently, Andrews established three partition theorems related to \(\mathcal {L}_d\) for \(d=1,2,3\) . Subsequently, Fu found the refinements of these three theorems by providing the bijective proofs. In this paper, in view of linked partition ideals, we give the analytic proofs of three refinements of the theorems given by Andrews. Meanwhile, we find an analogue of Euler pentagonal number theorem related to \(\mathcal {L}_d\) for \(d\geqslant 3\) . Furthermore, the combinatorial proofs of the main theorems are presented.
