On the divisibility of $q$-trinomial coefficients

TL;DR

This paper proves a congruence involving sums of central q-binomial coefficients, leading to new divisibility results for q-trinomial coefficients introduced by Andrews and Baxter.

Contribution

It introduces a novel q-congruence that implies divisibility properties of q-trinomial coefficients, expanding understanding of their algebraic structure.

Findings

Established a new q-congruence for sums of central q-binomial coefficients.

Derived divisibility properties for q-trinomial coefficients.

Enhanced the theoretical framework for q-series and combinatorial identities.

Abstract

We establish a congruence on sums of central $q$-binomial coefficients. From this $q$-congruence, we derive the divisibility of the $q$-trinomial coefficients introduced by Andrews and Baxter.