Polynomials whose coefficients are κ-fibonacci numbers

Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review

Abstract

Let {an}n≥0 denote the linear recursive sequence of order k (k ≥ 2) defined by the initial values a0 = a1 = · · · = ak-2 = 0 and ak-1 = 1 and the recursion an = an-1 + an-2 + · · · + an-k if n ≥ k. The an are often called k-Fibonacci numbers and reduce to the usual Fibonacci numbers when k = 2. Let Pn,k(x) = ak-1xn + akxn-1 + · · · + an+k-2x + an+k-1, which we will refer to as a k-Fibonacci coefficient polynomial. In this paper, we show for all k that the polynomial Pn,k(x) has no real zeros if n is even and exactly one real zero if n is odd. This generalizes the known result for the k = 2 and k = 3 cases corresponding to Fibonacci and Tribonacci coefficient polynomials, respectively. It also improves upon a previous upper bound of approximately k for the number of real zeros of Pn,k(x). Finally, we show for all k that the sequence of real zeros of the polynomials Pn,k(x) when n is odd converges to the opposite of the positive zero of the characteristic polynomial associated with the sequence an. This generalizes a previous result for the case k = 2.

Original languageEnglish
Pages (from-to)57-76
Number of pages20
JournalAnnales Mathematicae et Informaticae
Volume40
StatePublished - 2012

Keywords

  • K-fibonacci sequence
  • Linear recurrences
  • Zeros of polynomials

ASJC Scopus subject areas

  • Computer Science (all)
  • Mathematics (all)

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