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Polynomials whose coefficients are generalized Tribonacci numbers
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DOI：10.1016/j.amc.2012.12.052.png)
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Abstract
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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 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 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{1, 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 case p = q = r = 1, this convergence is monotonic. Finally, we are able to show the convergence in modulus of all the zeros of Q(n)(x) when p >= 1 >= q >= 0. (C) 2013 Elsevier Inc. All rights reserved.
Keywords:
Tribonacci numbers
Zeros of polynomials
Linear recurrences
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