Polynomial Identities Implying Capparelli's Partition Theorems

TL;DR

The paper develops polynomial identities that imply Capparelli's partition theorems, introduces new combinatorial interpretations, and derives novel identities involving q-trinomial coefficients and triple sums.

Contribution

It presents new polynomial identities, combinatorial interpretations, and q-trinomial coefficient identities that extend Capparelli's theorems and related results.

Findings

Polynomial identities implying Capparelli's partition theorems.

New combinatorial interpretations of identities.

Introduction of new q-trinomial coefficient identities.

Abstract

We propose and recursively prove polynomial identities which imply Capparelli's partition theorems. We also find perfect companions to the results of Andrews, and Alladi, Andrews and Gordon involving $q$-trinomial coefficients. We follow Kur\c{s}ung\"oz's ideas to provide direct combinatorial interpretations of some of our expressions. We use of the trinomial analogue of Bailey's lemma to derive new identities. These identities relate triple sums and products. A couple of new Slater type identities are also noted.