Polynomial sequences in groups

Given a group G with lower central series G = G(1) superset of or equal to G(2) superset of or equal to G(3) superset of or equal to ... ,We say that a sequence g: Z -> G is polynomial if for any k there is d such that the sequence obtained from g by applying the difference operator Dg(n) = g(n)(-1) g(n + 1) d times takes its values in G(k). We introduce the notion of the degree of a polynomial sequence and we prove that polynomial sequences of degrees not exceeding a given one form a group. As an application we obtain the following extension of the Hall-Petresco theorem: THEOREM. Let G = G(1) superset of or equal to G(2) superset of or equal to G(3) superset of or equal to ... be the lower central series of a group G. Let x is an element of G(k), y is an element of G(l) and let p, q be polynomials Z -> Z of degrees k and l, respectively. Then there is a sequence z(o) is an element of G, z(i) is an element of G(i) for i is an element of N, such that x(p(n))y(q(n)) [GRAPHICS] (C) 1998 Academic Press.