Exotic Bailey-Slater SPT-Functions II: Hecke-Rogers-Type Double Sums and Bailey Pairs From Groups A, C, E

TL;DR

This paper explores new spt-crank-type functions derived from Bailey pairs related to groups A, C, and E, providing new identities, congruences, and representations as double series and products, advancing partition theory and q-series analysis.

Contribution

It introduces four new spt-type functions from Bailey pairs of groups C and E, and establishes their identities, congruences, and representations using Bailey's Lemma and dissections.

Findings

Derived new spt-crank functions from Bailey pairs C1, C5, E2, E4.

Proved Ramanujan-type congruences for these functions.

Expressed functions as infinite products and Hecke-Rogers double series.

Abstract

We investigate spt-crank-type functions arising from Bailey pairs. We recall four spt-type functions corresponding to the Bailey pairs $A1$, $A3$, $A5$, and $A7$ of Slater and given four new spt-type functions corresponding to Bailey pairs $C1$, $C5$, $E2$, and $E4$. Each of these functions can be thought of as a count on the number of appearances of the smallest part in certain integer partitions. We prove simple Ramanujan type congruences for these functions that are explained by a spt-crank-type function. The spt-crank-type functions are two variable $q$-series determined by a Bailey pair, that when $z=1$ reduce to the spt-type functions. We find the spt-crank-type functions to have interesting representations as either infinite products or as Hecke-Rogers-type double series. These series reduce nicely when $z$ is a certain root of unity and allow us to deduce the congruences. Additionally we find dissections when $z$ is a certain root of unity to give another proof of the congruences. Our double sum and product formulas require Bailey's Lemma and conjugate Bailey pairs. Our dissection formulas follow from Bailey's Lemma and dissections of known ranks and cranks.