#### 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