Vol. 8, No. 2, 2015

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Convergence of the maximum zeros of a class of Fibonacci-type polynomials

Rebecca Grider and Kristi Karber

Vol. 8 (2015), No. 2, 211–220
Abstract

Let a be a positive integer and let k be an arbitrary, fixed positive integer. We define a generalized Fibonacci-type polynomial sequence by Gk,0(x) = a, Gk,1(x) = x a, and Gk,n(x) = xkGk,n1(x) + Gk,n2(x) for n 2. Let gk,n represent the maximum real zero of Gk,n. We prove that the sequence {gk,2n} is decreasing and converges to a real number βk. Moreover, we prove that the sequence {gk,2n+1} is increasing and converges to βk as well. We conclude by proving that {βk} is decreasing and converges to a.

Keywords
Fibonacci polynomial, convergence, zeros, roots
Mathematical Subject Classification 2010
Primary: 11B39
Secondary: 11B37, 30C15
Milestones
Received: 7 October 2012
Revised: 16 June 2013
Accepted: 19 October 2013
Published: 3 March 2015

Communicated by Kenneth S. Berenhaut
Authors
Rebecca Grider
Department of Mathematics
University of Oklahoma
601 Elm Avenue, Room 423
Norman, OK 73019
United States
Kristi Karber
Department of Mathematics and Statistics
University of Central Oklahoma
100 North University Drive
Edmond, OK 73034
United States