Open Access
Translator Disclaimer
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


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.


Download Citation

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.


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

Primary: 30C15

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

Rights: Copyright © 2015 Mathematical Sciences Publishers


Vol.8 • No. 1 • 2015
Back to Top