Polynomials whose coefficients are generalized Tribonacci numbers

Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review


Let an denote the third order linear recursive sequence defined by the initial values a0 = a1 = 0 and a2 = 1 and the recursion an = pan-1 + qan-2 + ra n-3 if n ≥ 3, where p, q, and r are constants. The an are generalized Tribonacci numbers and reduce to the usual Tribonacci numbers when p = q = r = 1 and to the 3-bonacci numbers when p = r = 1 and q = 0. Let Qn(X) = a2xn + a 3xn-1 +⋯+ an+1x + an+2, which we will refer to as a generalized Tribonacci coefficient polynomial. In this paper, we show that the polynomial Qn(X) has no real zeros if n is even and exactly one real zero if n is odd, under the assumption that p and q are non-negative real numbers with p ≥ max{l, q}. This generalizes the known result when p = q = r = 1 and seems to be new in the case when p = r = 1 and q = 0. Our argument when specialized to the former case provides an alternative proof of that result. We also show, under the same assumptions for p and q, that the sequence of real zeros of the polynomials Qn(x) when n is odd converges to the opposite of the positive zero of the characteristic polynomial associated with the sequence an. In the casep = q = r = 1, this convergence is monotonie. Finally, we are able to show the convergence in modulus of all the zeros of Qn(x) when p ≥ 1 ≥ q ≥ 0.

Original languageEnglish
Pages (from-to)8366-8374
Number of pages9
JournalApplied Mathematics and Computation
Issue number15
StatePublished - 1 Apr 2013


  • Linear recurrences
  • Tribonacci numbers
  • Zeros of polynomials

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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