Sums of entire functions having only real zeros

We show that certain sums of products of Hermite-Biehler entire functions have only real zeros, extending results of Cardon. As applications of this theorem we construct sums of exponential functions having only real zeros, we construct polynomials having zeros only on the unit circle, and we obtain...

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Bibliographic Details
Published in:arXiv.org 2007-08
Main Authors: Adams, Steven R, Cardon, David A
Format: Article
Language:English
Subjects:
Entire functions
Exponential functions
Mathematical analysis
Polynomials
Statistical mechanics
Sums
Theorems
Online Access:Get full text
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Summary:We show that certain sums of products of Hermite-Biehler entire functions have only real zeros, extending results of Cardon. As applications of this theorem we construct sums of exponential functions having only real zeros, we construct polynomials having zeros only on the unit circle, and we obtain the three-term recurrence relation for an arbitrary family of real orthogonal polynomials. We discuss a similarity of this result with the Lee-Yang circle theorem from statistical mechanics. Also, we state several open problems.
ISSN:2331-8422