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arXiv · 1312.2283

Non-Real Zero Decreasing Operators Related to Orthogonal Polynomials

Andre Bunton·Nicole Jacobs·Samantha Jenkins·Charles McKenry Jr.·Andrzej Piotrowski·Louis Scott
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Abstract

Laguerre's theorem regarding the number of non-real 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 Chebyshev, Legendre, and generalized Laguerre polynomial bases and (2) simultaneously generate a basis $B$ and a corresponding $B$-CZDS. Some extensions to transcendental entire functions in the Laguerre-Polya class are given which, in turn, give a new and short proof of a previously known result due to one of the authors. The paper concludes with several open questions.

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Andre Bunton, Nicole Jacobs, Samantha Jenkins, Charles McKenry Jr., Andrzej Piotrowski, Louis Scott. 2013-12-22. Non-Real Zero Decreasing Operators Related to Orthogonal Polynomials. https://doi.org/10.2140/involve.2015.8.129

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