Non-Real Zero Decreasing Operators Related to Orthogonal Polynomials

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.

15 pages, 0 figures, partial funding from NSA and NSF through an MAA NREUP grant

Cited by in corpus (2)

• Zeros of polynomials with four-term recurrence
• Parity of Polynomial Multiplier Sequences for the Chebyshev Basis