On a generalization of Schur theorem concerning resultants
type: Item
instance of: scholarly article
title: On a generalization of Schur theorem concerning resultants
zbMATH Open document ID: 1488.12003
author: Maciej Ulas
published in: Periodica Mathematica Hungarica
publication date: 2021-08-03T00:00:00Z
review text: Let \(K\) be a field, \(\mathbb{N}\) the set of non-negative integers, and \(\mathcal{A} := \{(i, j, k, m) \in \mathbb{N}^4: i \le j\) and \(m \le k\}\). For each \(A \in \mathcal{A}\), the paper under review considers the sequence \(r_{A,n}(x))_{n \in \mathbb{N}}\) of polynomials defined by the recurrence \[ r_{A,n}(x) = f_n(x)r_{A,n-1}(x) - v_nx^mr_{A,n-2}(x), \quad n \ge 2, \] where the initial polynomials \(r_{A,0}\), \(r_{A,1} \in K[x]\) are of degree \(i, j\), respectively, and \(f_n(x) \in K[x]\) is of degree \(n\) with variable coefficients, for any \(n \ge 2\). The paper under review finds a formula for the resultant \(\operatorname{Res}(r_{A,n}(x), r_{A,n-1}(x))\), which yields in the special case of \(A = (0, 1, 1, 0)\)) the classical Schur formula. The significance of the obtained formula is due to the fact (established by Favard) that each family of orthogonal polynomials corresponds to the sequence \((r_{A,n}(x))_{n \in \mathbb{N}}\), for suitably chosen sequences \((a_n)_{n \in \mathbb{N}}\), \((b_n)_{n \in \mathbb{N}}\) and \((c_n)_{n \in \mathbb{N}}\) (for a proof of Favard's theorem, see Theorem 4.4 of: \textit{T. S. Chihara} [An introduction to orthogonal polynomials. New York etc.: Gordon and Breach Science Publishers (1978; Zbl 0389.33008)]). As an application of the main result of the paper under review, the author gives the formula for the resultant \(\operatorname{Res}(r_{A,n}, r_{A,n-2})\), where \((r_{A,n}(x))_{n \in \mathbb{N}}\) is the sequence of orthogonal polynomials corresponding to a moment functional which is symmetric, i.e. its moments of odd order are zero. This condition is satisfied by the Legendre, Hermite, Chebyshev, Bessel, Lommel and many other sequences of orthogonal polynomials (see Chapter V of the above cited book). The author presents three illustrative examples which highlight the latter formula for the sequence of Legendre polynomials and the one of Hermite polynomials as well as for a certain sequence of polynomials …
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 12E10
Mathematics Subject Classification ID: 12E05
Mathematics Subject Classification ID: 13P15
zbMATH DE Number: 7377566
zbMATH Keywords: resultant
zbMATH Keywords: recurrence relations
zbMATH Keywords: orthogonal polynomials
MaRDI profile type: Publication
full work available at URL: s10998-020-00369-4
OpenAlex ID: W3109777071
cites work: Resultants of Cyclotomic Polynomials
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cites work: Discriminants of classical quasi-orthogonal polynomials with application to Diophantine equations
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DOI: 10.1007/S10998-020-00369-4
description: scientific article; zbMATH DE number 7377566