Asymmetric bilateral Bailey pairs and Rogers-Ramanujan type identities

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.