Generalized Taylor Series and Orders and Types of Entire Functions of Several Complex Variables
Fred Gross · Transactions of the American Mathematical Society · 1965
FRED GROSS1. Introduction.Entire functions of one complex variable which together with all their derivatives assume integer values at a finite number of integer points have been studied to a considerable extent.In particular certain lower bounds for the order and type of such transcendental integer valued entire functions have been obtained.Some studies related to this problem can be found in E. G.For entire functions of n complex variables, order and type are not numbers, but rather certain manifolds in R".The main purpose of this dissertation is to generalize some of the results obtained for entire functions of one complex variable to entire functions of several complex variables.For the sake of simplicity most of our theorems are stated for two variables.The generalizations to more variables are immediate.More specifically we would like to generalize some of the basic results of D. Sato [8] and E. G. Straus [1].We define a strongly transcendental function as a function /= 2Za¡jz[z2 such that a¡j # 0 for arbitrarily large i and /.We also call a transcendental function, which is not strongly transcendental, weakly transcendental.An entire function f(zx,z2) will be said to be Hurwitz if all its partial derivatives and its value are integral at the origin.We shall begin by proving a number of theorems and lemmas and by making some generalized definitions.These will enable us to study order and type points of entire functions in terms of their interpolation series (Generalized Taylor series) at a rectangular array of points.Two important facts that we shall generalize are the following theorems [8].I.There exist 2"° entire functions of order p, type a which together with all their derivatives assume integer values at the points 0, 1, 2,-,fc-1 for any order p with p ^ fc or p = fc and any type a with