Abstract:

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 $\phi=\frac{1+\sqrt{5}}{2}$.