Bilateral Bailey pairs and Rogers-Ramanujan type identities
Xiangxin Liu, Lisa Hui Sun
TL;DR
Using a bilateral Bailey framework, the paper constructs a key bilateral Bailey pair from the $q$-binomial theorem, $(\alpha_n(1,q), \beta_n(1,q)) = ((-1)^n z^n q^{\binom{n}{2}}, (z, q/z)_n /(q)_{2n})$, and employs bilateral lemmas, chains, and lattices to derive a broad family of Rogers–Ramanujan type identities, including unified parametric families such as $\sum_{n\ge0} \frac{(q^a, q^{m-a}; q^m)_n}{(q^m;q^m)_{2n}} q^{m n^2}$. By coupling this with Berkovich–McCoy–Schilling bilateral chains and Jouhet et al. bilateral lattices, it yields Appell–Lerch series identities and Andrews–Gordon type results, while Andrews–Warnaar bilateral lemmas produce identities for Hecke–type series. The work unifies many existing Rogers–Ramanujan type identities as special cases and expands the toolkit for connections between $q$-series, combinatorics, and mathematical physics. Overall, the paper provides a cohesive bilateral Bailey-based pipeline to generate and relate a wide spectrum of Rogers–Ramanujan type identities, Appell–Lerch identities, 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.