MORE SOLUTIONS TO VECTOR NONLINEAR RECURRENCE EQUATIONS

Let ℂ denote the complex numbers. Given a function F(z):ℂq→ℂq, suppose that w∈ℂq is a fixed point, that is, F(w)=w, and that F(z) is analytic at w. Then for 1≤p≤q, the q×1 vector recurrence equation zn+1=F(zn) for n=0,1,2,… ⁡, has a solution of the form zn=zn(w,αrn)= ∑i∈ℕp∞ai(w)(αrn)i, where a0p(w)=w, 0p=(0,… ⁡,0), α∈ℂp is arbitrary, (αrn)i= ∏k=1p(αkrkn)ik, and r1,… ⁡,rp are any distinct eigenvalues of Ḟ(w), where Ḟ(z)=dF(z)∕z′∈ℂq×q. The other ai(w) are given recursively using a new type of multivariate Bell polynomial.