Exotic Bailey-Slater SPT-Functions III: Bailey Pairs from Groups B, F, G, and J

TL;DR

This paper extends the study of spt-type functions derived from Bailey pairs, introducing new functions, proving Ramanujan-like congruences, and exploring their representations and dissections using Bailey's Lemma.

Contribution

It introduces new spt-type functions from Bailey pairs related to groups B, F, G, J, and establishes Ramanujan-type congruences with explanations via spt-crank-type functions.

Findings

Proved Ramanujan-type congruences for new spt-type functions.

Derived series and product representations of spt-crank-type functions.

Obtained dissections of spt-crank-type functions at roots of unity.

Abstract

We continue to investigate spt-type functions that arise from Bailey pairs. In this third paper on the subject, we proceed to introduce additional spt-type functions. We prove simple Ramanujan type congruences for these functions which can be explained by a spt-crank-type function. The spt-crank-type functions are actually defined first, with the spt-type functions coming from setting $z=1$ in the spt-crank-type functions. We find some of the spt-crank-type functions to have interesting representations as single series, some of which reduce to infinite products. Additionally we find dissections of the other spt-crank-type functions when $z$ is a certain root of unity. Both methods are used to explain congruences for the spt-type functions. Our series formulas require Bailey's Lemma and conjugate Bailey pairs. Our dissection formulas follow from Bailey's Lemma and dissections of known ranks and cranks.