Bailey pairs and quantum $q$-series identites. I. The classical identities

We use Bailey pairs to prove $q$-series identities at roots of unity due to Cohen and Bryson-Ono-Pitman-Rhoades. The proofs use Bailey pairs with quadratic forms developed in the study of mock theta functions. In addition to the standard Bailey lemma, we require some changes-of-base established by Bressoud-Ismail-Stanton. We then embed the identities in infinite families using the Bailey chain.