Andrews--Gordon and Stanton type identities: bijective and Bailey lemma approaches

Abstract

In 2018, Stanton proved two types of generalisations of the celebrated Andrews--Gordon and Bressoud identities (in their $q$-series version): one with a similar shape to the original identities, and one involving binomial coefficients. In this paper, we give new proofs of these identities. For the non-binomial identities, we give bijective proofs using the original Andrews--Gordon and Bressoud identities as key ingredients. These proofs are based on particle motion introduced by Warnaar and extended by the first and third authors and Konan. For the binomial identities, we use the Bailey lemma and key lemmas of McLaughlin and Lovejoy, and the order in which we apply the different lemmas plays a central role in the result. We also give an alternative proof of the non-binomial identities using the Bailey lattice. With each of these proofs, new Stanton-type generalisations of classical identities arise naturally, such as generalisations of Kur\c{s}ung\"oz's analogue of Bressoud's identity with opposite parity conditions, and of the Bressoud--G\"ollnitz--Gordon identities.