Article Contents
Article Contents

# A Favard type theorem for Hurwitz polynomials

• * Corresponding author: Abdon E. Choque-Rivero
• 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.

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

 Citation:

• 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

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

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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