Linear differential equations of infinite order and theta functions

Abstract

The purpose of this article is to show that some finiteness theorem (= finite dimensionality of the space of solutions) holds for a class of systems of linear differential equations of infinite order. Although finiteness theorems for holonomic systems of (micro-)differential equations of finite order have recently become quite popular, the character of the theorems which we present here is different from the results for equations of finite order. Hence, in this introduction, we discuss a simple and instructive example so that it may help the reader’s understanding of the character of the results in this article. As the example will indicate, our results have close connection with the celebrated result of Hamburger on the characterization of the c-function of Riemann, although we deal with theta functions (Hamburger [2], Hecke [3], and Weil [8]; see also Ehrenpreis and Kawai [ 11). This connection was pointed out to one of us (T.K.) by Professor L. Ehrenpreis. Concerning the basic properties of linear differential operators of infinite order, we refer the reader to Sato-Kawai-Kashiwara [6, Chap. II]’ (hereafter referred to as S-K-K). Here we only emphasize that a linear differential operator of infinite order acts upon the sheaf of holomorphic functions as a sheaf homomorphism. Hence our main result (Theorem 2.14 in Section 2) is of local character. This forms a striking contrast to the hitherto known way of characterizing theta functions through their automorphic properties. Now, in order to provide an example of our results, let us show how the theta zero-value (Nullwerte) is related to a system of linear differential equations of infinite order. In order to fix the notations, let us consider