Growth of polynilpotent varieties of Lie algebras and rapidly growing entire functions

We study the growth function $c_n(\mathbf V)$ for a variety of Lie algebras, where $c_n(\mathbf V)$ is the dimension of the linear hull of the set of multilinear words with $n$ different letters in the free algebra $F(\mathbf V,X)$ of the variety $\mathbf V$. With every non-trivial variety $\mathbf V$ of Lie algebras there is associated its complexity function $\mathscr C(\mathbf V,z)$, which is an entire function of a complex variable. In the case of a polynilpotent variety $\mathbf V$ of Lie algebras an estimate is obtained for the complexity function; in most cases it is of infinite order. We study the connection between the growth of a rapidly growing entire function and the asymptotics of its Taylor coefficients. The basic result is the asymptotics for the function $c_n(\mathbf V)$ in the case of a polynilpotent variety $\mathbf V$. Also, we prove an analogue of Regev's theorem for Lie algebras on upper estimates for the growth of arbitrary varieties. This gives more precision to the scale of superexponential growth of varieties of Lie algebras introduced earlier by the author.