## Abstract

Let a_{n} denote the third order linear recursive sequence defined by the initial values a_{0} = a_{1} = 0 and a_{2} = 1 and the recursion a_{n} = pa_{n-1} + qa_{n-2} + ra _{n}-_{3} if n ≥ 3, where p, q, and r are constants. The a_{n} 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 Q_{n}(X) = a_{2}x^{n} + a _{3}x^{n-1} +⋯+ a_{n+1}x + a_{n+2}, which we will refer to as a generalized Tribonacci coefficient polynomial. In this paper, we show that the polynomial Q_{n}(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 Q_{n}(x) when n is odd converges to the opposite of the positive zero of the characteristic polynomial associated with the sequence a_{n}. 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 Q_{n}(x) when p ≥ 1 ≥ q ≥ 0.

Original language | American English |
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Pages (from-to) | 8366-8374 |

Number of pages | 9 |

Journal | Applied Mathematics and Computation |

Volume | 219 |

Issue number | 15 |

DOIs | |

State | Published - 1 Apr 2013 |

## Keywords

- Linear recurrences
- Tribonacci numbers
- Zeros of polynomials

## All Science Journal Classification (ASJC) codes

- Computational Mathematics
- Applied Mathematics