Synopsis The differential operators in question are of the form G ( D Z ) where G ( w )is an entire function of order at most 1/ n and minimal type while D z 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 ( D Z ) 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 but also containing generalizations, some of which were later worked out by Klimczak. A basic tool in this paper is the characteristic series 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 D z are used to study the inversion problem for given F ( z ). In particular it is shown that exp ( D x )[ W ( z )] = 0 has the unique solution W ( z ) ≡ 0. Some singular boundary value problems are considered briefly.