Journal of Integer Sequences, Vol. 10 (2007), Article 07.6.8

Polynomials Generated by the Fibonacci Sequence

David Garth
Division of Mathematics & Computer Science
Truman State University
Kirksville, MO 63501

Donald Mills
Department of Mathematics
Rose-Hulman Institute of Technology
Terre Haute, IN 47803-3999

Patrick Mitchell
Department of Mathematics
Midwestern State University
Wichita Falls, TX 76308


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.

We answer several questions regarding these polynomials. Specifically, we show that each even-degree FCP has no real zeros, while each odd-degree FCP has a unique, and (for degree at least $3$) irrational, real zero. Further, we show that this sequence of unique real zeros converges monotonically to the negative of the golden ratio. Using Rouché's theorem, we prove that the zeros of the FCP's approach the golden ratio in modulus. We also prove a general result that gives the Mahler measures of an infinite subsequence of the FCP sequence whose coefficients are reduced modulo an integer $m\geq
2$. We then apply this to the case that $m=L_n$, the $n^{th}$ Lucas number, showing that the Mahler measure of the subsequence is $\phi^{n-1}$, where $\phi=\frac{1+\sqrt{5}}{2}$.

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(Concerned with sequences A000045 and A019523 .)

Received March 8 2007; revised version received June 11 2007. Published in Journal of Integer Sequences, June 19 2007.

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