Roots of a Type of Generalized Quasi-Fibonacci Polynomials

Let a be a nonnegative real number and define a quasi-Fibonacci polynomial sequence by F a 1 (x) = −a, F a 2 (x) = x − a, and F a n (x) = F a n−1(x) + xF a n−2(x) for n ≥ 2. Let ra n denote the maximum real root of F a n . We prove for certain values of a that the sequence {ra 2n} converges monotonically to βa = a 2 + a from above and the sequence {ra 2n+1} converges monotonically to βa from below.