arXiv 1805.024602018-05-07math.CVmath.CA

Geometry of limits of zeros of polynomial sequences of type $(1,1)$

David G. L. Wang, Jerry J. R. Zhang

备注：: 14 pages, 1 figure

英文摘要：

In this paper, 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 a "lollipop", or an interval. 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.

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