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

Abstract

Let {λn}n=1∞ be a sequence of distinct complex numbers diverging to infinity so that |λn|≤|λn+1| for all n∈N, and let {μn}n=1∞ be a sequence of positive integers. Consider the setΛ:={λ1,λ1,…,λ1︸μ1-times,λ2,λ2,…,λ2︸μ2-times,…,λk,λk,…,λk︸μk-times,…}. Subject to the condition μnlog⁡|λn|/|λn|→0 as n→∞, we prove that all non-entire Taylor–Dirichlet series of the form∑n=1∞(∑k=0μn−1cn,kzk)eλnz,cn,k∈C, 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.