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

Grider, Rebecca; Karber, Kristi

doi:10.2140/involve.2015.8.211

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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

DOI: 10.2140/involve.2015.8.211

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 $G_{k, 0} (x) = - a$ , $G_{k, 1} (x) = x - a$ , and $G_{k, n} (x) = x^{k} G_{k, n - 1} (x) + G_{k, n - 2} (x)$ for $n \geq 2$ . Let $g_{k, n}$ represent the maximum real zero of $G_{k, n}$ . We prove that the sequence ${g_{k, 2 n}}$ is decreasing and converges to a real number $β_{k}$ . Moreover, we prove that the sequence ${g_{k, 2 n + 1}}$ is increasing and converges to $β_{k}$ as well. We conclude by proving that ${β_{k}}$ 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

Vol. 8, No. 2, 2015