We study the root distribution of some univariate polynomials satisfying a recurrence of order two with linear polynomial coefficients. We show that the set of non-isolated limits of zeros of the polynomials is either an arc, or a circle, or an interval, or a “lollipop.” As an application, we discover a sufficient and necessary condition for the universal real-rootedness of the polynomials, subject to certain sign condition on the coefficients of the recurrence. Moreover, we obtain the sharp bound for all the zeros when they are real.

中文翻译：

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（1,1）型多项式序列的零点极限的几何

我们研究了一些具有线性多项式系数的二阶递归的单变量多项式的根分布。我们证明了多项式零的非孤立极限集是一个圆弧，一个圆，一个区间或一个“棒棒糖”。作为一种应用，我们发现了多项式的通用实根的充分必要条件，但要满足递归系数的某些符号条件。而且，当所有零为实数时，我们会获得它们的锐利边界。