By Pieter Cornelis. Sikkema
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Additional resources for Differential operators and differential equations of infinite order with constant coefficients: Researches in connection with integral functions of finite order
Now we use formula (21) if (16) holds, formula (22) if (17) holds and formula (23) if (18) holds. 44 In each of these three cases there exists to the number Z one and only one positive number t, which satisfies 1 (at) 4 = 1. aa J an J}%i = 00. 00 This means that I (x) possesses the property P 2. Consequently I (x) is an integral function of the order a and, because of (20), (24) and (P, 9) with e = a, r = t, it is an integral function of the normal type t of the order a. So, in case where (16) holds, the function I (x) is an integral function of the normal type of the order a.
Moreover, theorem 26 tells us that this function n(x) is even of the same kind as the function h(x). So 37 our theorem 26 is clearly an improvement on Whittaker's theorem I, in so far as in this theorem the integral function f(x) does not exceed the normal type of the order 1. To prove our theorem 26 we need a part of the results of our theorems 16, 17, 18, 19 and 21. These results needed we summarize in theorem 25 which, in fact, is a generalization of theorem 26. On account of the very general character of theorem 25 - as we know a certain class of "divergent" operators is also admitted, if the order of the given integral function h(x) is less than i - it is a far-going generalization of Whittaker's theorem I, in so far as in this theorem the integral function I (x) does not exceed the normal type of the order 1.
In each of the three cases that come up for discussion in theorems 8, 9 and 10 it appears, that the function h(x) = F(D) -* y(x) is an integral function of the same kind at most as the function y(x). ") If in this work we refer to a formula occurring in another chapter then that in question before the number of this formula we place the indication of that chapter. g. (II, 29) means formula (29) of chapter II; (P, 8) means formula (8) of the Preparatory Chapter. 28 In chapter III we restrict ourselves to those integral functions y(x) that do not exceed the minimum type of the order I and that are not identically equal to a constant.