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：

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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 \）和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 \）和\（P _ {-n}（z）= 0 \）。我们在\（a，\ b \）和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）\）是\（P_n（z）\）的零集。