On limit functions and their natural boundaries in infinite compositions of entire functions

Abstract

In this article, we consider infinite sequences {Φ n } of entire functions in the complex plane C defined as compositions of the form where each f n , n = 1, 2, … , is an entire function, and the limit functions Φ of such sequences. Under reasonable conditions on the sequence {f n }, and for the cases where Φ exists and is a nonconstant analytic function, one finds that the boundary of the domain where {Φ n } converges to Φ is in fact the natural boundary of Φ, and that this boundary satisfies certain “expansion” properties when considered under the composition of the f n 's. We also consider the case of constant limit functions Φ. In the final section we discuss the connection between the coefficients of a power series representation of a nonconstant limit Φ and the sequence {a n } of a one parameter family of entire functions f n (z) = f (a n,z ), whose composition as in (1) converges in some domain to Φ.