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On hyperbolic polynomials with four-term recurrence and linear coefficients
Calcolo ( IF 1.4 ) Pub Date : 2020-08-04 , DOI: 10.1007/s10092-020-00373-7
Richard Adams

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 \(P_n(z)\).

中文翻译:

具有四项递归和线性系数的双曲多项式

对于任何实数\(a,\ b \)c,我们形成满足四项递归$的多项式序列\(\ {P_n(z)\} _ {n = 0} ^ \ infty \)$ \ begin {aligned} P_n(z)+ azP_ {n-1}(z)+ bP_ {n-2}(z)+ czP_ {n-3}(z)= 0,\ n \ in {\ mathbb {N}},\ end {align} $$与初始条件\(P_0(z)= 1 \)\(P _ {-n}(z)= 0 \)。我们在\(a,\ b \)c上找到了必要和充分的条件,在这些条件下\(P_n(z)\)的零对于所有n都是真实的,并提供了一个明确的真实区间,其中\(\ bigcup \ nolimits _ {n = 0} ^ \ infty {\ mathcal {Z}}(P_n)\)是密集的,其中\({\ mathcal {Z}}(P_n)\)\(P_n(z)\)的零集。
更新日期:2020-08-04
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