Exotic Bailey-Slater SPT-Functions I: Group A

TL;DR

This paper introduces new spt-type functions derived from Bailey pairs, proves Ramanujan-type congruences for them, and constructs associated crank-type functions, expanding the understanding of partition-related functions and their symmetries.

Contribution

It presents novel spt-type functions from Bailey pairs, establishes Ramanujan-type congruences, and develops crank-type functions with root of unity dissections, advancing partition theory.

Findings

Proved Ramanujan-type congruences for new spt-functions

Constructed spt-crank-type functions with root of unity dissections

Utilized Chan's identity on generalized Lambert series

Abstract

We introduce several spt-type functions that arise from Bailey pairs. 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 constructed by adding an extra variable $z$ into the generating functions. We find dissections when $z$ is a certain root of unity, as has been done for many rank and crank difference formulas of various partition type objects. Our formulas require an identity of Chan on generalized Lambert series.