An Andrews–Gordon type identity for overpartitions

Abstract

In 1961, Gordon found a combinatorial generalization of the Rogers–Ramanujan 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.