Representation of entire functions

Abstract

In this chapter we consider the representation of entire functions by series of generalized Appell polynomials. First we shall see how the class of functions that can be represented, and the number of expansions of a given function, depend on properties of the functions A, Ψ and g. Then we shall study the effect of various specializations and finally we shall illustrate some points of the theory by means of particular sets of polynomials. We suppose throughout the chapter that the function Ψ of (6.1) is a comparison function (§ 2), and hence necessarily entire. We suppose also that A(w) and g(w) are regular at 0. We may then choose a region Ω _w in the w-plane in which A(w) is regular and g(w) is regular and univalent [since we supposed g’(0)≠0]. If ϱ ₀ is the distance from the origin to the nearest point of the boundary of Ω _w , the series (6.1) is convergent for all w in the open disk Δ _w : |w|<ϱ ₀. Let ζ = g(w) map Ω _w onto a set Ω _ζ in the ζ-plane, and denote the image of Δ _w by Δ _ζ < Ω _ζ . Let the inverse of g be w = W(ζ), and set B(ζ) = A(W(ζ)). Then

$$ B\left( \zeta \right)\psi \left( {z\zeta } \right) = \sum\limits_{{n = 0}}^{\infty } {{p_{n}}\left( z \right)} {\left\{ {W\left( \zeta \right)} \right\}^{n}} $$

((7.1))

, with the series converging uniformly in compact subsets of Δ _ζ . A set of generalized Appell polynomials is often defined by a generating relation of the form (7.1) instead of (6.1).