Growth of finitely generated polynilpotent Lie algebras and groups, generalized partitions, and functions analytic in the unit circle

Recently, the author has suggested a series of dimensions of algebras which includes as first terms dimension of a vector space, Gelfand-Kirillov dimension, and superdimension. These dimensions enabled us to describe the change of the growth in the transition from a Lie algebra to its universal enveloping algebra. In fact, this is a result on some generalized partitions. In this paper, we obtain more precise asymptotics for generalized partitions. As a main application, we obtain more precise asymptotics for the growth of free polynilpotent finitely generated Lie algebras. As a corollary, we specify the asymptotic growth of the lower central series ranks for free polynilpotent finitely generated groups. We essentially use Hilbert-Poincare series and some facts on the growth of functions analytic in the unit circle. By the growth of such functions, we mean their growth when the variable tends to 1. Finally, we study two kinds of p-central series for free polynilpotent finitely generated groups. We obtain asymptotics for the ranks of these series, in one case we have an example of a polynomial, but not rational growth.