Some Lucas-type congruences for q-trinomial coefficients

Abstract: In this paper, we present several new $q$-congruences on the $q$-trinomial coefficients introduced by Andrews and Baxter. As a conclusion, we obtain the following congruence: \begin{align*} \bigg(!!\binom{ap+b}{cp+d}!!\bigg)\equiv\bigg(!!\binom{a}{c}!!\bigg)\bigg(!!\binom{b}{d}!!\bigg)+\bigg(!!\binom{a}{c+1}!!\bigg)\bigg(!!\binom{b}{d-p}!!\bigg)\pmod{p}, \end{align*} where $a,b,c,d$ are integers subject to $a \geq 0, 0 \leq b,d \leq p-1$, and $p$ is an odd prime. Besides, we find that the method can also be used to reprove Pan's Lucas-type congruence for the $q$-Delannoy numbers.