On some resultants formulas of Schur type

Let $(r_{A,n}(x){)}_{n\in \mathbb{N}}$ be a sequence of polynomials with coefficients from a field $K$ satisfying the recurrence relation $r_{A,n}(x)= \sum_{|\alpha|\leq m} t_{\alpha,n}(x)$\mathbf{\text{r}}${A,n}^\alpha(x)$oforder$d+1 \in $\mathbb{N}${+}$,where$t_{\alpha,n} \in K[x]$,$m \in $\mathbb{N}${+}$ are fixed, $\alpha \in {\mathbb{N}}^{d+1}$, $\mathrm{\mid }\alpha \mathrm{\mid }={\alpha }_0+\dots +{\alpha }_d$ and $$\mathbf{\text{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 $r_{A,0},\dots ,r_{A,d}$ 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 $d=1$.