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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
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Abstract |
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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 and a corresponding complex zero decreasing sequence for the basis . 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
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Mathematical Subject Classification 2010
Primary: 30C15
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Milestones
Received: 22 December 2013
Revised: 29 March 2014
Accepted: 7 April 2014
Published: 10 December 2014
Communicated by Michael Dorff
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Authors |
| Andre Bunton | |
| University of Alaska Southeast 11120 Glacier Highway Juneau, AK 99801 United States |
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| Nicole Jacobs | |
| University of Alaska Southeast 11120 Glacier Highway Juneau, AK 99801 United States |
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| Samantha Jenkins | |
| University of Alaska Southeast 11120 Glacier Highway Juneau, AK 99801 United States |
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| Charles McKenry Jr. | |
| University of Alaska Southeast 11120 Glacier Highway Juneau, AK 99801 United States |
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| Andrzej Piotrowski | |
| University of Alaska Southeast 11120 Glacier Highway Mail Stop SOB1 Juneau, 99801 United States |
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| Louis Scott | |
| University of Alaska Southeast 11120 Glacier Highway Juneau, AK 99801 United States |
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