## Involve: A Journal of Mathematics

• Involve
• Volume 8, Number 2 (2015), 211-220.

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

#### 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 $Gk,n(x) = xkGk,n−1(x) + Gk,n−2(x)$ 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 $a$.

#### Article information

Source
Involve, Volume 8, Number 2 (2015), 211-220.

Dates
Revised: 16 June 2013
Accepted: 19 October 2013
First available in Project Euclid: 22 November 2017

https://projecteuclid.org/euclid.involve/1511370856

Digital Object Identifier
doi:10.2140/involve.2015.8.211

Mathematical Reviews number (MathSciNet)
MR3320854

Zentralblatt MATH identifier
1310.11022

#### Citation

Grider, Rebecca; Karber, Kristi. Convergence of the maximum zeros of a class of Fibonacci-type polynomials. Involve 8 (2015), no. 2, 211--220. doi:10.2140/involve.2015.8.211. https://projecteuclid.org/euclid.involve/1511370856

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