#### 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 ${G}_{k,0}\left(x\right)=-a$, ${G}_{k,1}\left(x\right)=x-a$, and ${G}_{k,n}\left(x\right)={x}^{k}{G}_{k,n-1}\left(x\right)+{G}_{k,n-2}\left(x\right)$ for $n\ge 2$. Let ${g}_{k,n}$ represent the maximum real zero of ${G}_{k,n}$. We prove that the sequence $\left\{{g}_{k,2n}\right\}$ is decreasing and converges to a real number ${\beta }_{k}$. Moreover, we prove that the sequence $\left\{{g}_{k,2n+1}\right\}$ is increasing and converges to ${\beta }_{k}$ as well. We conclude by proving that $\left\{{\beta }_{k}\right\}$ 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 