Polynomials whose coefficients are κ-fibonacci numbers

Let {a_n}_n≥0 denote the linear recursive sequence of order k (k ≥ 2) defined by the initial values a₀ = a₁ = · · · = a_k-2 = 0 and a_k-1 = 1 and the recursion a_n = a_n-1 + a_n-2 + · · · + a_n-k if n ≥ k. The an are often called k-Fibonacci numbers and reduce to the usual Fibonacci numbers when k = 2. Let P_n,k(x) = a_k-1xⁿ + a_kx_n-1 + · · · + a_n+k-2^x + a_n+k-1, which we will refer to as a k-Fibonacci coefficient polynomial. In this paper, we show for all k that the polynomial P_n,k(x) has no real zeros if n is even and exactly one real zero if n is odd. This generalizes the known result for the k = 2 and k = 3 cases corresponding to Fibonacci and Tribonacci coefficient polynomials, respectively. It also improves upon a previous upper bound of approximately k for the number of real zeros of P_n,k(x). Finally, we show for all k that the sequence of real zeros of the polynomials P_n,k(x) when n is odd converges to the opposite of the positive zero of the characteristic polynomial associated with the sequence an. This generalizes a previous result for the case k = 2.