Bilateral Bailey Lattices and Andrews-Gordon Type Identities

TL;DR

This paper extends the Bailey lattice to a bilateral form, providing new proofs and identities related to Andrews-Gordon and Bressoud identities, enriching the combinatorial and q-series theory.

Contribution

It introduces a simple bilateral Bailey lattice and derives new identities and proofs, expanding the scope of Bailey lattice applications.

Findings

Bilateral Bailey lattice extension from the bilateral Bailey lemma

New m-versions of Andrews-Gordon and Bressoud identities

Elementary proof of a general Bressoud identity

Abstract

We show that the Bailey lattice can be extended to a bilateral version in just a few lines from the bilateral Bailey lemma, using a very simple lemma transforming bilateral Bailey pairs relative to $a$ into bilateral Bailey pairs relative to $a/q$. Using this and similar lemmas, we give bilateral versions and simple proofs of other (new and known) Bailey lattices, including a Bailey lattice of Warnaar and the inverses of Bailey lattices of Lovejoy. As consequences of our bilateral point of view, we derive new $m$-versions of the Andrews-Gordon identities, Bressoud's identities, a new companion to Bressoud's identities, and the Bressoud-G\"ollnitz-Gordon identities. Finally, we give a new elementary proof of another very general identity of Bressoud using one of our Bailey lattices.