Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Differential Operators of infinite order

Einar Hillea1

a1 La Jolla, California, U.S.A.

Synopsis

The differential operators in question are of the form G(DZ) where G(w)is an entire function of order at most 1/n and minimal type while Dz is a linear differential operator of order n with coefficients which are entire ( = integral) functions of z, usually polynomials. This class of operators form a natural generalization of the class G(d/dz) studied during the first half of the century Muggli, Polya, Ritt and others. The class G(DZ) was introduced by the present author and his pupils in the 1940s. In fact, the present paper is partly based on a MS from that period, mostly devoted to the special case

S0308210500012026_eqnU1

but also containing generalizations, some of which were later worked out by Klimczak. A basic tool in this paper is the characteristic series

S0308210500012026_eqnU2

Examples are given showing that the domain of absolute convergence of such a series need neither be convex nor of finite connectivity, a question which has puzzled the author for forty odd years. Characteristic series arising from regular or singular boundary value problems for the operator Dz are used to study the inversion problem

S0308210500012026_eqnU3

for given F(z). In particular it is shown that exp (Dx)[W(z)] = 0 has the unique solution W(z) ≡ 0. Some singular boundary value problems are considered briefly.

(Received August 07 1979)