New infinite hierarchies of polynomial identities related to the Capparelli partition theorems

TL;DR

This paper introduces new polynomial identities related to Capparelli's theorems, using Bailey's lemma to generate infinite families of sum-product identities and exploring duality transformations in partition theory.

Contribution

It provides a novel polynomial refinement of Capparelli's identities and develops infinite hierarchies of identities through Bailey's lemma, expanding the understanding of partition identities.

Findings

Established new polynomial refinements of Capparelli's identities

Derived infinite families of sum-product identities from finite analogues

Explored $q o 1/q$ duality and related partition relations

Abstract

We prove a new polynomial refinement of the Capparelli's identities. Using a special case of Bailey's lemma we prove many infinite families of sum-product identities that root from our finite analogues of Capparelli's identities. We also discuss the $q\mapsto 1/q$ duality transformation of the base identities and some related partition theoretic relations.