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On a generalization of Schur theorem concerning resultants
Periodica Mathematica Hungarica ( IF 0.8 ) Pub Date : 2020-11-23 , DOI: 10.1007/s10998-020-00369-4
Maciej Ulas

Let K be a field and put $${\mathcal {A}}:=\{(i,j,k,m)\in \mathbb {N}^{4}:\;i\le j\;\text{ and }\;m\le k\}$$ . For any given $$A\in {\mathcal {A}}$$ we consider the sequence of polynomials $$(r_{A,n}(x))_{n\in \mathbb {N}}$$ defined by the recurrence $$\begin{aligned} r_{A,n}(x)=f_{n}(x)r_{A,n-1}(x)-v_{n}x^{m}r_{A,n-2}(x),\;n\ge 2, \end{aligned}$$ where the initial polynomials $$r_{A,0}, r_{A,1}\in K[x]$$ are of degree i, j respectively and $$f_{n}\in K[x], n\ge 2$$ , is of degree k with variable coefficients. The aim of the paper is to prove the formula for the resultant $${\text {Res}}(r_{A,n}(x),r_{A,n-1}(x))$$ . Our result is an extension of the classical Schur formula which is obtained for $$A=(0,1,1,0)$$ . As an application we get the formula for the resultant $${\text {Res}}(r_{A,n},r_{A,n-2})$$ , where the sequence $$(r_{A,n})_{n\in \mathbb {N}}$$ is the sequence of orthogonal polynomials corresponding to a moment functional which is symmetric.

中文翻译:

关于结果的 Schur 定理的推广

设 K 为一个域,并把 $${\mathcal {A}}:=\{(i,j,k,m)\in \mathbb {N}^{4}:\;i\le j\;\ text{ 和 }\;m\le k\}$$ 。对于任何给定的 $$A\in {\mathcal {A}}$$ 我们考虑多项式序列 $$(r_{A,n}(x))_{n\in \mathbb {N}}$$通过重复 $$\begin{aligned} r_{A,n}(x)=f_{n}(x)r_{A,n-1}(x)-v_{n}x^{m}r_{ A,n-2}(x),\;n\ge 2, \end{aligned}$$ 其中初始多项式 $$r_{A,0}, r_{A,1}\in K[x]$ $ 分别为 i, j 度, $$f_{n}\in K[x], n\ge 2$$ 为 k 度,具有可变系数。本文的目的是证明结果 $${\text {Res}}(r_{A,n}(x),r_{A,n-1}(x))$$ 的公式。我们的结果是对 $$A=(0,1,1,0)$$ 获得的经典 Schur 公式的扩展。作为应用程序,我们得到结果 $${\text {Res}}(r_{A,n},r_{A,n-2})$$ 的公式,其中序列 $$(r_{A,
更新日期:2020-11-23
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