Elementary Polynomial Identities Involving $q$-Trinomial Coefficients

TL;DR

This paper proves new polynomial identities involving q-trinomial coefficients using the q-binomial theorem, connects them to Capparelli's partition theorems, and proposes an infinite hierarchy of identities.

Contribution

It introduces three novel polynomial identities with q-trinomial coefficients and links them to partition theorems, expanding the theoretical framework.

Findings

Derived three new polynomial identities involving q-trinomial coefficients.

Connected identities to Capparelli's partition theorems in the limit.

Proposed an infinite hierarchy of polynomial identities.

Abstract

We use $q$-binomial theorem to prove three new polynomial identities involving $q$-trinomial coefficients. We then use summation formulas for the $q$-trinomial coefficients to convert our identities into another set of three polynomial identities, which imply Capparelli's partition theorems when the degree of the polynomial tends to infinity. This way we also obtain an interesting new result for the sum of the Capparelli's products. We finish this paper by proposing an infinite hierarchy of polynomial identities.