  Let $(r_{A,n}(x))_{n \in \mathbb{N}}$ be a sequence of polynomials with
coefficients from a field $K$ satisfying the recurrence relation $r_{A,n}(x)=
\sum_{|\alpha|\leq m} t_{\alpha,n}(x)\textbf{r}_{A,n}^\alpha(x)$ of order $d+1
\in \mathbb{N}_{+}$, where $t_{\alpha,n} \in K[x]$, $m \in \mathbb{N}_{+}$ are
fixed, $\alpha \in \mathbb{N}^{d+1}$, $|\alpha| = \alpha_0 + \ldots+\alpha_d$
and
$\textbf{r}_{A,n}^\alpha(x)=r_{A,n-1}^{\alpha_0}(x)r_{A,n-2}^{\alpha_1}(x)\cdots
r_{A,n-d-1}^{\alpha_d}(x).$ We show that under mild assumptions on the initial
polynomials $r_{A,0}, \ldots, r_{A,d}$ and the coefficients $t_{\alpha,n}$, we
can give the expression for the resultant $\text{Res}(r_{A,n}, r_{A,n-1})$. Our
results generalize recent result of Ulas concerning the case $m=1$ and $d=1$.
