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

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 ${\mathrm{\Phi }}_n(q)$ for $(a,b)=(m,m-1)$.