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

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.