Staircases to analytic sum-sides for many new integer partition identities of Rogers-Ramanujan type

TL;DR

This paper develops a novel technique using staircases and jagged partitions to derive analytic sum-sides for various Rogers-Ramanujan type partition identities, including new conjectures and generalizations.

Contribution

It introduces a new method for obtaining analytic sum-sides, conjectures new identities related to affine Lie algebra modules, and generalizes existing identities such as Capparelli's.

Findings

New analytic sum-sides for Rogers-Ramanujan type identities

Conjectured identities linked to affine Lie algebra modules

Generalizations of Capparelli identities

Abstract

We utilize the technique of staircases and jagged partitions to provide analytic sum-sides to some old and new partition identities of Rogers-Ramanujan type. Firstly, we conjecture a class of new partition identities related to the principally specialized characters of certain level $2$ modules for the affine Lie algebra $A_9^{(2)}$. Secondly, we provide analytic sum-sides to some earlier conjectures of the authors. Next, we use these analytic sum-sides to discover a number of further generalizations. Lastly, we apply this technique to the well-known Capparelli identities and present analytic sum-sides which we believe to be new. All of the new conjectures presented in this article are supported by a strong mathematical evidence.