2015/03/10 by J. L. Gross, Gross, J. L., T. Mansour +5 · 1 citation
Mathematics · #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.1503.05404
openalex publication_date 2015/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the sequence of polynomials Wn(x) defined by the recursion Wn(x)=(ax+b)Wn-1(x)+dWn-2(x), with initial values W0(x)=1 and W1(x)=t(x-r), where a,b,d,t,r are real numbers, a,t>0, and d<0. We show that every polynomial Wn(x) is distinct-real-rooted, and that the roots of the polynomial Wn(x) interlace the roots of the polynomial Wn-1(x). We find that, as n→∞, the sequence of smallest roots of the polynomials Wn(x) converges decreasingly to a real number, and that the sequence of largest roots converges increasingly to a real number. Moreover, by using the Dirichlet approximation theorem, we prove that there is a number to which, for every positive integer i≥2, the sequence of ith smallest roots of the polynomials Wn(x) converges. Similarly, there is a number to which, for every positive integer i≥2, the sequence of ith largest roots of the polynomials Wn(x) converges. It turns out that these two convergence points are independent of the numbers t and r, as well as i. We derive explicit expressions for these four limit points, and we determine completely when some of these limit points coincide.