Bilateral Bailey pairs and Rogers-Ramanujan type identities

TL;DR

This paper introduces bilateral Bailey pairs derived from the q-binomial theorem and uses bilateral Bailey lemmas, chains, and lattices to derive and unify various Rogers-Ramanujan type identities, including new identities involving Appell-Lerch series and Hecke-type series.

Contribution

The paper develops a bilateral Bailey pair framework and applies bilateral Bailey lemmas, chains, and lattices to derive and unify a wide class of Rogers-Ramanujan type identities, extending previous results.

Findings

Derived new bilateral Bailey pairs from the q-binomial theorem.

Unified many Rogers-Ramanujan type identities using bilateral Bailey methods.

Obtained identities involving Appell-Lerch series and Hecke-type series.

Abstract

Rogers-Ramanujan type identities occur in various branches of mathematics and physics. As a classic and powerful tool to deal with Rogers-Ramanujan type identities, the theory of Bailey's lemma has been extensively studied and generalized. In this paper, we found a bilateral Bailey pair that naturally arises from the q-binomial theorem. By applying the bilateral versions of Bailey lemmas, Bailey chains and Bailey lattices, we derive a number of Rogers-Ramanujan type identities, which unify many known identities as special cases. Further combined with the bilateral Bailey chains due to Berkovich, McCoy and Schilling and the bilateral Bailey lattices due to Jouhet et al., we also obtain identities on Appell-Lerch series and identities of Andrews-Gordon type. Moreover, by applying Andrews and Warnaar's bilateral Bailey lemmas, we derive identities on Hecke-type series.