A Favard type theorem for Hurwitz polynomials

Abdon E. Choque-Rivero; Iván Area

doi:10.3934/dcdsb.2019252

Article Contents

2020, Volume 25, Issue 2: 529-544. Doi: 10.3934/dcdsb.2019252

This issue Previous Article Existence and exponential stability for neutral stochastic integro–differential equations with impulses driven by a Rosenblatt process Next Article Nonnegative oscillations for a class of differential equations without uniqueness: A variational approach

A Favard type theorem for Hurwitz polynomials

Abdon E. Choque-Rivero^1, , and
Iván Area^2,

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
^* Corresponding author: Abdon E. Choque-Rivero

Received: February 2019

Revised: April 2019

Early access: November 2019

Published: February 2020

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Abstract

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.

Keywords:

Hurwitz polynomials,
orthogonal polynomials,
Markov moments,
three-term recurrence relation coefficients.

Mathematics Subject Classification: Primary: 12D10, 26C10, 33C45, 34K20, 39A30.

Citation:

Full Text(HTML)

Figure 1. 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

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Figure 2. 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

Download: Full-size image PowerPoint slide

Figure 3. 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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