Polynomials whose coefficients are generalized Tribonacci numbers

摘要

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+ran-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+a3xn-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{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 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 case p=q=r=1, this convergence is monotonic. Finally, we are able to show the convergence in modulus of all the zeros of Qn(x) when p⩾1⩾q⩾0.