Zeta Functions and Green's Functions for Linear Partial Differential Operators of Elliptic Type with Constant Coefficients

Abstract

in which T(nj) is an arbitrary polynomial of elliptic (will be defined) type of any even degree, and Q, is any further polynomial and we will demonstrate that these series really have sums, for any real variables xj and all complex s; they they are Cx and even analytic in the xj, and entire functions in s; except that for (xj) $ (0)-and of course for congruent points-there are some simple poles in the s-plane of the kind familiar from the theory of Zeta functions, Epstein Zeta functions, and such like expansions. There are no equations present however, at least not in an analytically fruitful manner, unless one chooses to interpret Theorem 11 as embodying a functional equation incipiently; and some remarks tending to substantiate such an interpretation for Theorem 11 have been presented in a separate note, see [1}. Actually, in our theorem proper, T(nj) and also Q(nj) will not be polynomials at all, but rather more general functions of non-oscillatory behavior reminiscent of such, and it will be syllogistically advisable to have these generalizations undertaken in the body of the statements, but we will not attempt to assess the gains from such generalizations in terms of concretely novel consequences thus resulting. If with a linear operator with constant coefficients h ,~~~~~~~~rl+ * +rkf (3) Af = (1) E a,, ...rk 1 . a~k O<rl+. +rk < 2h OX1 * an