An Andrews–Gordon type identity for overpartitions

Abstract

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.

Congruences modulo $4$ for Rogers--Ramanujan--Gordon type overpartitions

“…In 2015, Chen, Sang and Shi [9] obtained the overpartition analogue of Theorem 1.1 in the case j = 0. The generating function form was also derived by Sang and Shi [14]. Here we state these two theorems in a unified form together with its generating function form.…”

Bressoud style identities for regular partitions and overpartitions

“…In other words, some of the partition counters in Definition 7 have their generating functions expressed as multiple sums. Known examples are the Andrews-Gordon identities themselves [3], their counterpart for Bressoud's all-moduli generalization [6], Andrews-Gordon identities for overpartitions [7], and their Bressoud style generalization for overpartitions [19]. These constitute multiple series generating functions for 1 B 0 k,a (n), 2 B 0 k,a (n), 1 B 0 k,a (n), and 2 B 0 k,a (n).…”

The Bressoud-Göllnitz-Gordon Theorem for Overpartitions of Even Moduli

“…It has been proved in [11] that the first reduction operation of p-th kind and The first reduction dilation of p-th kind are mutually inverse mapping. What is more, it has been proved in [30] that such two operation preserve the three conditions of G N 1 ,...,N k−1 ;i . Now we aim to give a proof of Theorem 3.3.…”

“…Let E k,i (m, n) denote the number of overpartitions counted by E k,i (n) with exactly m parts, Sang and Shi [30] gave a combinatorial proof of the following generating function of E k,i (m, n).…”