Root geometry of polynomial sequences III: Type (1,1) with positive coefficients

2017/12/17 by David G. L. Wang, Jiarui Zhang, Wang, David G. L. +1
Mathematics · Physics and Astronomy · #03D20 #26C10 #30C15 #Advanced Differential Equations and Dynamical Systems #Advanced Mathematical Theories #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1712.06105

openalex publication_date 2017/12/17 · openalex created_date 2018/01/05 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the root distribution of some univariate polynomials W_n(z) satisfying a recurrence of order two with linear polynomial coefficients over positive numbers. We discover a sufficient and necessary condition for the overall real-rootedness of all the polynomials, in terms of the polynomial coefficients of the recurrence. Moreover, in the real-rooted case, we find the set of limits of zeros, which turns out to be the union of a closed interval and one or two isolated points; when non-real-rooted polynomial exists, we present a sufficient condition under which every polynomial with n large has a real zero.

Citations

• Common zeros of polynomials satisfying a recurrence of order two

Related

• Geometry of limits of zeros of polynomial sequences of type (1,1)
• Root geometry of polynomial sequences I: Type (0,1)
• Geometry of limits of zeros of polynomial sequences of type (1,2)
• Root geometry of polynomial sequences II: Type (1,0)
• Common zeros of polynomials satisfying a recurrence of order two