#### Vol. 8, No. 1, 2015

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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
Primary: 30C15