Some Smallest Parts Functions from Variations of Bailey's Lemma

TL;DR

This paper introduces new smallest parts partition and crank functions derived from Bailey's Lemma variations, satisfying simple congruences and connecting to hypergeometric functions, advancing partition theory.

Contribution

It constructs novel smallest parts functions and identities using Bailey's Lemma variations, linking to hypergeometric functions and congruences.

Findings

Functions satisfy simple linear congruences modulo 3 and 5

Introduces identities for four-variable q-hypergeometric functions

Connects new functions to known spt-crank-type functions

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.