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 $G_{k,0}\left(x\right)=-a$, $G_{k,1}\left(x\right)=x-a$, 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 $a$.