Two new Bailey pairs and their $q$-identities of Rogers-Ramanujan type modulo 15, 24, and 30

TL;DR

This paper introduces two new Bailey pairs derived from generalizations of Euler's pentagonal number theorem, leading to new Rogers-Ramanujan type $q$-identities modulo 15, 24, and 30.

Contribution

The paper presents novel Bailey pairs and applies them to derive new $q$-series identities of Rogers-Ramanujan type for specific moduli.

Findings

New Bailey pairs related to Euler's pentagonal number theorem

Derived $q$-identities of Rogers-Ramanujan type for mod 15, 24, 30

Established $q$-series transformations using the new Bailey pairs

Abstract

In this paper, we first establish two new Bailey pairs via finding two generalizations of Euler's pentagonal number theorem. Next, we specificize the Bailey lemmas with these two Bailey pairs. As applications, we finally establish some $q$-series transformations and $q$-identities of Rogers-Ramanujan type modulo 15, 24, and 30.