Some Smallest Parts Functions from Variations of Bailey's Lemma

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.