On Fibonacci-type polynomial recurrences of order two and the accumulation points of their set of zeros

Batra, Prashant (2018) On Fibonacci-type polynomial recurrences of order two and the accumulation points of their set of zeros. Annales Mathematicae et Informaticae, 49. pp. 33-41. ISSN 1787-6117

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Abstract

We identify the accumulation points of the zero set of the polynomial family Gn+1(z) := zGn(z) + Gn−1(z), n 2 N, generated from complex polynomial seeds G0,G1. This problem has been treated recently, for seed pairings of constants with linear polynomials, by Böttcher and Kittaneh (2016). We determine the accumulation points in the general case of arbitrary co-prime polynomial seeds, thus simplifying and streamlining previous approaches. Keywords: Fibonacci polynomials; three-term recurrences; zero attractor; asymptotic zero location. MSC: Primary: 11B39. Secondary: 30C15; 30B15; 40A15.

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