Bilateral Bailey Lattices and Andrews-Gordon Type Identities
Jehanne Dousse, Frédéric Jouhet, Isaac Konan
TL;DR
This work generalizes the Bailey lattice to a bilateral setting, introducing simple transformation lemmas that map bilateral Bailey pairs from relative to $a$ to $a/q$ and enabling bilateral lattices and $N$-extensions. The bilateral framework unifies and extends existing lattices (Warnaar, Lovejoy) and yields new $m$-versions of Andrews–Gordon, Bressoud and related identities, including a new elementary proof of a general Bressoud identity. The authors develop recurrence machinery and two key bilateral lemmas to prove a broad bilateral $N$-growth of Bailey lattices and to derive numerous identities, culminating in bilateral proofs of Br80 variants and new $m$-versions of Göllnitz–Gordon-type identities. The results broaden the analytic toolkit for $q$-series, with potential implications for partition theory and connections to higher-rank and bilinear structures in combinatorics and representation theory.
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-Göllnitz-Gordon identities. Finally, we give a new elementary proof of another very general identity of Bressoud using one of our Bailey lattices.