SOLUTIONS TO NONLINEAR RECURRENCE EQUATIONS

Let F(z) be any function. Suppose that w is a fixed point of F(z), that is, F(w) = w. Then the recurrence equationxn+1 = F(xn)for n = 0, 1, 2, ... has a solution of the formxn(w) = w + P infinity i=1 ai 1AiF.1(w)in,where F.1(z) = d F(z)/dz. So, for each w there is a set of complex x0 such that x0(w) = x0. We assume that F(z) is analytic at w. This solution appears to be new, even for such famous examples like the logistic map and the Mandelbrot equation.