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2015 Nonreal zero decreasing operators related to orthogonal polynomials
Andre Bunton, Nicole Jacobs, Samantha Jenkins, Charles McKenry, Andrzej Piotrowski, Louis Scott
Involve 8(1): 129-146 (2015). DOI: 10.2140/involve.2015.8.129

Abstract

Laguerre’s theorem regarding the number of nonreal zeros of a polynomial and its image under certain linear operators is generalized. This generalization is then used to (1) exhibit a number of previously undiscovered complex zero decreasing sequences for the Jacobi, ultraspherical, Legendre, Chebyshev, and generalized Laguerre polynomial bases and (2) simultaneously generate a basis B and a corresponding complex zero decreasing sequence for the basis B. An extension to transcendental entire functions in the Laguerre–Pólya class is given, which, in turn, gives a new and short proof of a previously known result due to Piotrowski. The paper concludes with several open questions.

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Andre Bunton. Nicole Jacobs. Samantha Jenkins. Charles McKenry. Andrzej Piotrowski. Louis Scott. "Nonreal zero decreasing operators related to orthogonal polynomials." Involve 8 (1) 129 - 146, 2015. https://doi.org/10.2140/involve.2015.8.129

Information

Received: 22 December 2013; Revised: 29 March 2014; Accepted: 7 April 2014; Published: 2015
First available in Project Euclid: 22 November 2017

zbMATH: 1311.30002
MathSciNet: MR3321716
Digital Object Identifier: 10.2140/involve.2015.8.129

Subjects:
Primary: 30C15

Keywords: complex zero decreasing sequences , diagonalizable linear operators , orthogonal polynomials , zeros of polynomials

Rights: Copyright © 2015 Mathematical Sciences Publishers

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