Fibonacci-mandelbrot polynomials and matrices

Abstract

We explore a family of polynomials similar to the Mandelbrot polynomials called the Fibonacci-Mandelbrot polynomials defined by q₀(z) = 0, q₁(z) = 1, and q_n(z) = zq_n−1q_n−2 + 1. We compute the roots of the Fibonacci-Mandelbrot polynomials using two methods. One method uses a recursively constructed matrix, where elements are 0, 1, or −1, whose eigenvalues are the roots of q_n(z). The other method uses a special-purpose homotopy continuation method, where the solution of the differential equation, [EQUATION], in which the initial condition are 0, and the roots of q_n−1 and q_n−2, are also the roots of the Fibonacci-Mandelbrot polynomials.