A representation formula of sum rules for zeros of polynomials
type: Item
instance of: scholarly article
title: A representation formula of sum rules for zeros of polynomials
zbMATH Open document ID: 0768.34054
DOI: 10.1007/BF02844663
publication date: 1993-02-03T00:00:00Z
review text: Let \(P_ N(x) = \prod^ N_{k=1}(x-x_ k)\) be an eigenfunction of the linear differential operator \(\sum^ n_{k=0}g_ k(x)f^{(k)}\), where \(g_ k(x)\) is a polynomial of degree not higher than \(k\), and suppose that all zeros of \(P_ N(x)\) are simple. The author obtains an explicit recurrence relation for the quantities \(y_ s = \sum^ N_{i=1} x^ s_ i\) by representing the so-called sum rules \(J_ s^{(i)}\), defined by \[ J_ s^{(i)} = \sum{x^ s_{\ell_ 1}\over (x_{\ell_ 1} - x_{\ell_ 2})(x_{\ell_ 1} - x_{\ell_ 3})\dots(x_{\ell_ 1} - x_{\ell_ i})}, \] in terms of the \(y_ t\) \((t \leq s-i + 1)\). The representation formula is derived using Lucas polynomials.
Mathematics Subject Classification ID: 34L05
Mathematics Subject Classification ID: 26C10
Mathematics Subject Classification ID: 33C70
zbMATH DE Number: 109282
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: eigenfunction
zbMATH Keywords: linear differential operator
zbMATH Keywords: sum rules
zbMATH Keywords: Lucas polynomials
author: P. E. Ricci
reviewed by: Erik A. van Doorn
MaRDI profile type: Publication
cites work: Sum rules for zeros of polynomials. I
cites work: On the polynomial solutions of ordinary differential equations of the fourth order
cites work: On the zeros of eigenfunctions of polynomial differential operators
cites work: Solution of the Difference Equations of Generalized Lucas Polynomials
cites work: An Explicit Formula for $f(\mathcal{A})$ and the Generating Functions of the Generalized Lucas Polynomials
cites work: Q3671491
full work available at URL: https://doi.org/10.1007/bf02844663
OpenAlex ID: W2048555621
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 109282