Polynomials whose coefficients are generalized Tribonacci numbers

Let a_n denote the third order linear recursive sequence defined by the initial values a₀ = a₁ = 0 and a₂ = 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) = a₂xⁿ + a ₃x^n-1 +⋯+ a_n+1x + 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.