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A Favard type theorem for Hurwitz polynomials

  • * Corresponding author: Abdon E. Choque-Rivero

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

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  • 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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