Representation of functions that are regular at the origin

Abstract

In Chapter II, the function Ψ used in defining a set of generalized Appell polynomials was itself an entire function; the functions for which we obtained expansions were in the class ℝ _Ψ , which is a class of entire functions. In this chapter we take up the more complicated case in which Ψ is merely regular in some neighborhood of the origin. We suppose again that $\psi (t) = \sum\limits_{n = 0}^0 {{\psi _n}} {t^n}$ with Ψ _n > 0, we suppose that lim Ψ _n ^1/n exists (finite), and we suppose, by way of normalization, that thie limit is 1. Then Ψ(t) is regular for all t in a set E that contains the open disk |t| <1 and has t=1 as a boundary point. In Chapter II ws singled out for study the class ℝ _Ψ , of functions f(z) = ∑f _n z ⁿ of finite Ψ-type, i.e. such that lim sup |f _n /Ψ _n |^1/n<∞. The notion of finite Ψ-type is no longer of much use: the relation lim sup |f _n /Ψ _n |^1/n= R now merely asserts that f is analytic in the disk |z| < 1/R, regardless of the choice of Ψ, so that ℝ _Ψ is the set of all functions f that are regular at the origin. We find it more useful to study the class A(Ω) consisting of all functions which are analytic in a specified simply-connected neighborhood Ω of the origin. Certain fundamental differences are at once apparent between the present situation and that studied before. In Chapter II, an expansion that represented a given entire function represented it everywhere.