We present properties of the classic Fibonacci-type recursive polynomials defined by Hoggatt in 1973 and the Golden-type recursive polynomials described by Moore in 1993. This work introduces a second order recursive sequence of Golden-like polynomials defined by \(\displaystyle G_{n+1}(x) = x^{k}G_{n}(x) + x^{l}G_{n-1}(x), \;k,l\;\mbox{positive integers} \) with \(G_0=-1,\;G_1=x-1\) In this work, we explore some properties of this Golden-type polynomial sequence. We derive a Binet form for the polynomial, as well as a matrix representation. Several properties of the sequence of the maximum real roots will also be presented to explain the analytic behavior of \(G_n\) .

错误:搜索内容不能为空,请输入英文关键词
错误:关键词超出字数限制,请精简
高级检索

Some Remarks on Fibonacci-type Recursive Polynomials

  • Kristen Hallas,
  • Joan Mattle,
  • Deanna Perez,
  • Aklilu Zeleke

摘要

We present properties of the classic Fibonacci-type recursive polynomials defined by Hoggatt in 1973 and the Golden-type recursive polynomials described by Moore in 1993. This work introduces a second order recursive sequence of Golden-like polynomials defined by \(\displaystyle G_{n+1}(x) = x^{k}G_{n}(x) + x^{l}G_{n-1}(x), \;k,l\;\mbox{positive integers} \) with \(G_0=-1,\;G_1=x-1\) In this work, we explore some properties of this Golden-type polynomial sequence. We derive a Binet form for the polynomial, as well as a matrix representation. Several properties of the sequence of the maximum real roots will also be presented to explain the analytic behavior of \(G_n\) .