Some Smallest Parts Functions from Variations of Bailey's Lemma

Abstract: We construct new smallest parts partition functions and smallest parts crank functions by considering variations of Bailey's Lemma and conjugate Bailey pairs. The functions we introduce satisfy simple linear congruences modulo $3$ and $5$. We introduce and give identities for two four variable $q$-hypergeometric functions; these functions specialize to some of our new spt-crank-type functions as well as many known spt-crank-type functions.