Solutions of infinite order differential equations without the grouping phenomenon and a generalization of the Fabry-Pólya theorem

Let {λn}n=1∞ be a sequence of distinct complex numbers diverging to infinity so that |λ_n|≤|λ_n+1| for all n∈N, and let {μn}n=1∞ be a sequence of positive integers. Consider the set Subject to the condition μ_nlog|λ_n|/|λ_n|→0 as n→∞, we prove that all non-entire Taylor-Dirichlet series of the form have a convex natural boundary if and only if Λ is an interpolating variety for the space of entire functions of infraexponential type A|z|0. Our result is in the spirit of the Fabry-Pólya gap results.We also prove that if Λ is the zero set of some F∈A|z|0 but not an interpolating variety, it is still possible for the solutions of the differential equation of infinite order F(d/dz) f= 0 to admit a Taylor-Dirichlet series representation, that is, a representation without groupings.