Andrews--Gordon type identities with parity restrictions through particle motion

TL;DR

This paper extends Andrews--Gordon type identities with parity restrictions using particle motion bijections, providing new proofs and generalizations of existing q-series identities and related algebraic structures.

Contribution

It introduces new q-series identities with parity restrictions and generalizes previous identities using a particle motion bijection approach.

Findings

Proved multisum equals sum of products identities.

Generalized Andrews and Kim--Yee identities.

Provided a simple proof of a recent algebraic identity.

Abstract

In this paper, we use the particle motion bijection introduced by Warnaar and developed by the two authors, Jouhet and Konan, to study q-series and partition identities of the Andrews--Gordon type with parity restrictions. These restrictions are of the type ``even (resp. odd) parts appear an even number of times". We prove $q$-series identities where a multisum equals a sum of products, which generalise identities of Andrews and Kim--Yee in a similar way that Stanton's identities generalised the Andrews--Gordon identities. As a consequence of our results, we obtain a simple proof of a recent identity of Chern--Li--Stanton--Xue--Yee related to Ariki--Koike algebras.