Polynomials generated by the Fibonacci sequence

Garth, David ; Mills, Donald ; Mitchell, Patrick

Journal of integer sequences, Tome 10 (2007) no. 6

Résumé

The Fibonacci sequence's initial terms are $F_{0}=0$ and $F_{1}=1$, with $F_{n}=F_{n-1}+F_{n-2}$ for $n \geq 2$. We define the polynomial sequence ${\bf p}$ by setting $p_{0}(x)=1$ and $p_{n}(x)=xp_{n-1}(x)+F_{n+1}$ for $n \geq 1$, with $p_{n}(x)=\sum_{k=0}^{n}F_{k+1}x^{n-k}$. We call $p_{n}(x)$ the Fibonacci-coefficient polynomial (FCP) of order $n$. The FCP sequence is distinct from the well-known Fibonacci polynomial sequence.

Détail

Keywords: Fibonacci, sequence, polynomial, zero, root, rouché's theorem, mahler measure