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01.09.2020

On hyperbolic polynomials with four-term recurrence and linear coefficients

verfasst von: Richard Adams

Erschienen in: Calcolo | Ausgabe 3/2020

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Abstract

For any real numbers $a,\ b$, and c, we form the sequence of polynomials $\{P_n(z)\}_{n=0}^\infty$ satisfying the four-term recurrence

$$\begin{aligned} P_n(z)+azP_{n-1}(z)+bP_{n-2}(z)+czP_{n-3}(z)=0,\ n\in {\mathbb {N}}, \end{aligned}$$

with the initial conditions $P_0(z)=1$ and $P_{-n}(z)=0$. We find necessary and sufficient conditions on $a,\ b$, and c under which the zeros of $P_n(z)$ are real for all n, and provide an explicit real interval on which $\bigcup \nolimits _{n=0}^\infty {\mathcal {Z}}(P_n)$ is dense, where ${\mathcal {Z}}(P_n)$ is the set of zeros of $P_n(z)$.

Vorheriger Artikel Shape preserving -fractal rational cubic splines

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Titel: On hyperbolic polynomials with four-term recurrence and linear coefficients
verfasst von: Richard Adams
Publikationsdatum: 01.09.2020
Verlag: Springer International Publishing
Erschienen in: Calcolo / Ausgabe 3/2020
Print ISSN: 0008-0624
Elektronische ISSN: 1126-5434
DOI: https://doi.org/10.1007/s10092-020-00373-7

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