Refined $q$-Trinomial Coefficients and Two Infinite Hierarchies of $q$-Series Identities

TL;DR

This paper proves a new identity involving refined q-trinomial coefficients and extends it to two infinite hierarchies of q-series identities, one of which relates to Capparelli's Partition Theorem.

Contribution

It introduces a novel identity with refined q-trinomials and develops two infinite hierarchies of identities using iterative transformations.

Findings

Proved a new identity involving refined q-trinomial coefficients.

Extended the identity to two infinite hierarchies of q-series identities.

One hierarchy includes an identity equivalent to Capparelli's first Partition Theorem.

Abstract

We will prove an identity involving refined $q$-trinomial coefficients. We then extend this identity to two infinite families of doubly bounded polynomial identities using transformation properties of the refined $q$-trinomials in an iterative fashion in the spirit of Bailey chains. One of these two hierarchies contains an identity which is equivalent to Capparelli's first Partition Theorem.