Generalizing Results on the Convergence of the Maximum Roots of Fibonacci Type Polynomials

摘要

Let k be a positive integer and define a Fibonacci-type polynomial sequence by G_0(x) = - 1, G_1(x) = x - 1, and G_n{x) = x~kG_(n-1)(x) + G_(n_2)(x) for n ≥ 2. Let g_n denote the maximum real root of G_n. We prove that there exists a real number α_k such that the sequence {g_(2n)} converges monotonically to α_k from above and the sequence {g_(2n+1)} converges monotonically to α_k from below. We also prove that {α_k} converges monotonically to 1 from above.