Further results on the divisibility of $q$-trinomial coefficients

TL;DR

This paper investigates the divisibility properties of certain $q$-trinomial coefficients introduced by Andrews and Baxter, providing complete results for specific cases modulo the square of cyclotomic polynomials.

Contribution

It precisely determines the divisibility of $q$-trinomial coefficients $ au_0$, $T_0$, and $T_1$ for particular parameter choices modulo $\

Findings

Complete divisibility results for $ au_0$, $T_0$, and $T_1$ when parameters are multiples of $n$.

Explicit characterization of these coefficients modulo $\

The results extend understanding of divisibility properties of $q$-trinomial coefficients.

Abstract

We study divisibility for the $q$-trinomial coefficients $\tau_0(n,m,q)$, $T_0(n,m,q)$ and $T_1(n,m,q)$, which were first introduced by Andrews and Baxter. In particular, we completely determine $\tau_0(an,bn,q)$, $T_0(an,bn,q)$ and $T_1(an,bn,q)$ modulo the square of the cyclotomic polynomial $\Phi_n(q)$ for $(a,b)=(m,m-1)$.