Let B k,i (n) be the number of partitions of n with certain difference condition and let A k,i (n) be the number of partitions of n with certain congruence condition. The Rogers-Ramanujan-Gordon theorem states that B k,i (n) = A k,i (n). Lovejoy obtained an overpartition analogue of the Rogers-Ramanujan-Gordon theorem for the cases i = 1 and i = k. We find an overpartition analogue of the Rogers-Ramanujan-Gordon theorem in the general case. Let D k,i (n) be the number of overpartitions of n satisfying certain difference condition and C k,i (n) be the number of overpartitions of n whose non-overlined parts satisfy certain congruence condition. We show that C k,i (n) = D k,i (n). By using a function introduced by Andrews, we obtain a recurrence relation that implies that the generating function of D k,i (n) equals the generating function of C k,i (n). By introducing the Gordon marking of an overpartition, we find a generating function formula for D k,i (n) that can be considered an overpartition analogue of an identity of Andrews for ordinary partitions.

For k ≥ i ≥ 1, let B k,i (n) denote the number of partitions of n such that part 1 appears at most i − 1 times, two consecutive integers l and l + 1 appear at most k − 1 times and if l and l + 1 appear exactly k − 1 times then the total sum of the parts l and l + 1 is congruent to i − 1 modulo 2. Let A k,i (n) denote the number of partitions with parts not congruent to i, 2k − i and 2k modulo 2k. Bressoud's theorem states that A k,i (n) = B k,i (n). Corteel, Lovejoy, and Mallet found an overpartition analogue of Bressoud's theorem for i = 1, that is, for partitions not containing nonoverlined part 1. We obtain an overpartition analogue of Bressoud's theorem in the general case. For k ≥ i ≥ 1, let D k,i (n) denote the number of overpartitions of n such that the nonoverlined part 1 appears at most i−1 times, for any integer l, l and nonoverlined l + 1 appear at most k − 1 times and if the parts l and the nonoverlined part l + 1 appear exactly k − 1 times then the total sum of the parts l and nonoverlined part l + 1 is congruent to the number of overlined parts that are less than l + 1 plus i − 1 modulo 2. Let C k,i (n) denote the number of overpartitions with the nonoverlined parts not congruent to ±i and 2k − 1 modulo 2k − 1. We show that C k,i (n) = D k,i (n). This relation can also be considered as a Rogers-Ramanujan-Gordon type theorem for overpartitions.

We show that the number of anti-lecture hall compositions of n with the first entry not exceeding k − 2 equals the number of overpartitions of n with non-overlined parts not congruent to 0, ±1 modulo k. This identity can be considered as a finite version of the anti-lecture hall theorem of Corteel and Savage. To prove this result, we find two Rogers-Ramanujan type identities for overpartitions which are analogous to the Rogers-Ramanujan type identities due to Andrews. When k is odd, we give another proof by using the bijections of Corteel and Savage for the anti-lecture hall theorem and the generalized Rogers-Ramanujan identity also due to Andrews.

In 1961, Gordon found a combinatorial generalization of the RogersRamanujan identities, which has been called the Rogers-Ramanujan-Gordon theorem. In 1974, Andrews derived an identity which can be considered as the generating function counterpart of the Rogers-Ramanujan-Gordon theorem, and it has been called the Andrews-Gordon identity. The Andrews-Gordon identity is an analytic generalization of the Rogers-Ramanujan identities with odd moduli. In 1979, Bressoud obtained a Rogers-Ramanujan-Gordon type theorem and the corresponding Andrews-Gordon type identity with even moduli. In 2003, Lovejoy proved two overpartition analogues of two special cases of the Rogers-Ramanujan-Gordon theorem. In 2013, Chen, Sang and Shi found the overpartition analogue of the Rogers-Ramanujan-Gordon theorem in general cases and the corresponding Andrews-Gordon type identity with even moduli. In 2008, Corteel, Lovejoy and Mallet found an overpartition analogue of a special case of Bressoud's theorem of the Rogers-Ramanujan-Gordon type. In 2012, Chen, Sang and Shi obtained the overpartition analogue of Bressoud's theorem in the general case. In this paper, we obtain an Andrews-Gordon type identity corresponding to this overpartition theorem with odd moduli using the Gordon marking representation of an overpartition.

In this paper, we find an identity which connects the overpartition function and the function of Rogers-Ramanujan-Gordon type overpartitions by considering the weights and gaps. This identity can be seen as an analogue of the weighted identity of Alladi.