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February  2020, 25(2): 529-544. doi: 10.3934/dcdsb.2019252

## A Favard type theorem for Hurwitz polynomials

 1 Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Ciudad Universitaria, C.P. 58048, Morelia, Michoacán, México 2 Departamento de Matemática Aplicada Ⅱ, E.E. Aeronáutica e do Espazo, Universidade de Vigo, 32004-Ourense, Spain

* Corresponding author: Abdon E. Choque-Rivero

Dedicated to Prof. Juan J. Nieto on the occasion of his 60th birthday

Received  February 2019 Revised  April 2019 Published  November 2019

A Favard type theorem for Hurwitz polynomials is proposed. This result is a sufficient condition for a sequence of polynomials of increasing degree to be a sequence of Hurwitz polynomials. As in the Favard celebrated theorem, the three-term recurrence relation is used. Some examples of Hurwitz sequences are also presented. Additionally, a characterization of constructing a family of orthogonal polynomials on $[0, \infty)$ by two couples of numerical sequences $({A_{1, j}, B_{1, j}})$ and $({A_{2, j}, B_{2, j}})$ is stated.

Citation: Abdon E. Choque-Rivero, Iván Area. A Favard type theorem for Hurwitz polynomials. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 529-544. doi: 10.3934/dcdsb.2019252
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##### References:
Zeros of the polynomials $f_{n}(z)$ for $\alpha = \sqrt{2}$ and $n = 1, 2, 3, 4, 5$: $f_{1}(z)$ in black, $f_{2}(z)$ in blue, $f_{3}(z)$ in magenta, $f_{4}(z)$ in orange, and $f_{5}(z)$ in red
Zeros of the polynomials $f_{n}(z)$ for $n = 1, 2, 3, 4, 5$: $f_{1}(z)$ in black color, $f_{2}(z)$ in blue, $f_{3}(z)$ in magenta, $f_{4}(z)$ in orange, and $f_{5}(z)$ in red
Zeros of the polynomials $f_{n}(z)$ for $\alpha = 3$ and $n = 1, 2, 3, 4, 5$: $f_{1}(z)$ in black, $f_{2}(z)$ in blue, $f_{3}(z)$ in magenta, $f_{4}(z)$ in orange, and $f_{5}(z)$ in red
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