Polynomial identities implying Capparelli's partition theorems

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 Kurşungöz's ideas to provide direct combinatorial interpretations of some of our expressions. We make use of the trinomial analogue of Bailey's lemma to derive new identities. These identities relate certain triple sums and products. A couple of new Slater type identities involving bases q2, q3, q6, and q12 are also proven. We also discuss a new infinite hierarchy containing these Slater type identities.