Asymmetric bilateral Bailey pairs and Rogers-Ramanujan type identities

TL;DR

This paper introduces asymmetric bilateral Bailey pairs and lemmas, deriving new Rogers-Ramanujan type identities and identities involving false theta functions and Appell-Lerch series.

Contribution

It extends Bailey pair theory to asymmetric bilateral cases, leading to novel identities and a new Bailey lemma based on the Bailey lattice.

Findings

Derived two asymmetric bilateral Bailey pairs.

Obtained identities of Rogers-Ramanujan, Andrews-Gordon, and false theta functions.

Established an asymmetric bilateral Bailey lemma related to Appell-Lerch series.

Abstract

The theory of Bailey's transform provides a systematic method for deriving $q$-identities, the key factor of which is the Bailey pair. The concept of Bailey pair was first extended to bilateral version by Paule. In this paper, following Rogers' work on Fourier series, we derive two asymmetric bilateral Bailey pairs. By inserting them into the bilateral Bailey chains, we obtain several identities of Rogers-Ramanujan type, Andrews-Gordon type and also identities on false theta functions. Furthermore, based on the Bailey lattice due to Dousse, Jouhet and Konan, we get an asymmetric bilateral Bailey lemma which leads to identities on Appell-Lerch series. Moreover, by using the asymmetric bilateral Bailey lemmas due to Andrews and Warnaar, we get some identities on false theta functions and the generalized Hecke-type series.