This paper investigates the zero distribution of a sequence of polynomials \(\left\{ P_{m}(z)\right\} _{m=0}^{\infty }\) which satisfy a four-term recurrence whose coefficients are linear polynomials in z. In particular, we study necessary and sufficient conditions for the reality of the zeros of \(P_{m}(z)\). Under these conditions, we find an explicit interval containing these zeros, whose union forms a dense subset of this interval.

中文翻译：

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具有四项递归和线性系数的多项式的零点

本文研究了满足四项递归系数的多项式\（\ left \ {P_ {m}（z）\ right \} _ {m = 0} ^ {\ infty} \）的零分布是z中的线性多项式。特别地，我们研究了\（P_ {m}（z）\）的零的存在的充要条件。在这些条件下，我们找到一个包含这些零的明确间隔，其并集构成该间隔的密集子集。