On Monotonicity of Some Sequences Related to Hyperfibonacci Numbers and Hyperlucas Numbers!

Abstract

In this paper, we mainly discuss the monotonicity of some sequences related to the hyperfibonacci sequences $\{F_n^{[r]}{\}}_{n\ge 0}$ and the hyperlucas sequences $\{L_n^{[r]}{\}}_{n\ge 0}$, where $r$ is a positive integer. We prove that $\{\sqrt[n]{F_n^{[1]}}{\}}_{n\ge 1}$ and $\{\sqrt[n]{F_n^{[2]}}{\}}_{n\ge 1}$ are unimodal and $\{\sqrt[n]{L_n^{[1]}}{\}}_{n\ge 1}$, $\{\sqrt[n]{F_{n+1}^{[1]}/F_n^{[1]}}{\}}_{n\ge 1}$, and $\{\sqrt[n]{L_{n+1}^{[1]}/L_n^{[1]}}{\}}_{n\ge 2}$ are decreasing. Furthermore, we discuss the monotonicity of the sequences

$${\left\{\frac{\sqrt[n+1]{F_{n+1}^{[1]}}}{\sqrt[n]{F_n^{[1]}}}\right\}}_{n\ge 1}\text{ and }{\left\{\frac{\sqrt[n+1]{L_{n+1}^{[1]}}}{\sqrt[n]{L_n^{[1]}}}\right\}}_{n\ge 1}$$