Overpartitions and Bressoud's conjecture, I

Abstract

In a memoir in 1980, Bressoud obtained an analytic generalization of the Rogers-Ramanujan-Gordon identities based on Andrews' generalization of Watson's q-analogue of Whipple's theorem. Let j = 0 or 1, Bressoud also defined two partition functions A j and B j depending on multiple parameters as combinatorial counterparts of his identity, where the function A j can be viewed as the generating function of partitions with certain congruence conditions and the function B j can be viewed as the generating function of partitions with certain difference conditions. Bressoud conjectured A j = B j which specializes to most of the well-known theorems in the theory of partitions, including the Rogers-Ramanujan-Gordon identities and the Andrews-Göllnitz-Gordon identities. Special cases of his conjecture have been subsequently proved by Bressoud, Andrews, Kim and Yee. Recently, Kim proved that Bressoud's conjecture is true for the general case when j = 1.In this paper, we introduce a new partition function B j which could be viewed as an overpartition analogue of the partition function B j introduced by Bressoud. By constructing a bijection, we showed that there is a relationship between B 1 and B 0 and a relationship between B 0 and B 1 . Based on the relationship between B 1 and B 0 and Bressoud's theorems on the Rogers-Ramanujan-Gordon identities and the Göllnitz-Gordon identities, we obtain the overpartition analogue of the Rogers-Ramanujan-Gordon identities due to Chen, Sang and Shi and a new overpartition analogue of the Andrews-Göllnitz-Gordon identities. On the other hand, by using the relation between B 0 and B 1 and Bressoud's conjecture for j = 1 proved by Kim, we obtain an overpartition analogue of Bressoud's conjecture for j = 0, which provides overpartition analogues of many classical partition theorems including Euler's partition theorem. The generating function of the overpartition analogue of Bressoud's conjecture for j = 0 is also obtained with the aid of Bailey pairs.