Vol. 8, No. 1, 2015

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ISSN: 1944-4184 (e-only)
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Nonreal zero decreasing operators related to orthogonal polynomials

Andre Bunton, Nicole Jacobs, Samantha Jenkins, Charles McKenry Jr., Andrzej Piotrowski and Louis Scott

Vol. 8 (2015), No. 1, 129–146
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.

Keywords
complex zero decreasing sequences, diagonalizable linear operators, zeros of polynomials, orthogonal polynomials
Mathematical Subject Classification 2010
Primary: 30C15
Milestones
Received: 22 December 2013
Revised: 29 March 2014
Accepted: 7 April 2014
Published: 10 December 2014

Communicated by Michael Dorff
Authors
Andre Bunton
University of Alaska Southeast
11120 Glacier Highway
Juneau, AK 99801
United States
Nicole Jacobs
University of Alaska Southeast
11120 Glacier Highway
Juneau, AK 99801
United States
Samantha Jenkins
University of Alaska Southeast
11120 Glacier Highway
Juneau, AK 99801
United States
Charles McKenry Jr.
University of Alaska Southeast
11120 Glacier Highway
Juneau, AK 99801
United States
Andrzej Piotrowski
University of Alaska Southeast
11120 Glacier Highway
Mail Stop SOB1
Juneau, 99801
United States
Louis Scott
University of Alaska Southeast
11120 Glacier Highway
Juneau, AK 99801
United States