Non-Real Zero Decreasing Operators Related to Orthogonal Polynomials
Complex Variables
2016-01-20 v2
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 and a corresponding -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.
Cite
@article{arxiv.1312.2283,
title = {Non-Real Zero Decreasing Operators Related to Orthogonal Polynomials},
author = {Andre Bunton and Nicole Jacobs and Samantha Jenkins and Charles McKenry and Andrzej Piotrowski and Louis Scott},
journal= {arXiv preprint arXiv:1312.2283},
year = {2016}
}
Comments
15 pages, 0 figures, partial funding from NSA and NSF through an MAA NREUP grant