Polynomials whose coefficients are k-Fibonacci numbers

Let {a(n)}(n >= 0) denote the linear recursive sequence of order k (k >= 2) defined by the initial values a(0) = a(1) = center dot center dot center dot = a(k-2) = 0 and a(k-1) = 1 and the recursion a(n) = a(n-1) + a(n-2) + center dot center dot center dot + a(n-k) if n >= k. The a(n) are often called k-Fibonacci numbers and reduce to the usual Fibonacci numbers when k = 2. Let P-n,P- k(x) = a(k-1)x(n) + a(k)x(n-1) + center dot center dot center dot + 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,P- 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,P- k(x). Finally, we show for all k that the sequence of real zeros of the polynomials P-n,P- 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.