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

Division of Mathematics & Computer Science

Truman State University

Kirksville, MO 63501

USA

Donald Mills

Department of Mathematics

Rose-Hulman Institute of Technology

Terre Haute, IN 47803-3999

USA

Patrick Mitchell

Department of Mathematics

Midwestern State University

Wichita Falls, TX 76308

USA

**Abstract:**

The Fibonacci sequence's initial terms are and ,
with
for . We define the polynomial
sequence by setting and
for , with
. We call the *Fibonacci-coefficient polynomial (FCP) of order *. 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 ) 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 . We then apply this to the case that , the Lucas number, showing that the Mahler measure of the subsequence is , where .

(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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