On some resultants formulas of Schur type

Abstract

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 $d=1$.