Representation of finite order solutions to linear differential equations with exponential sum coefficients

TL;DR

This paper characterizes when linear differential equations with exponential sum coefficients have finite order entire solutions, providing explicit forms of these solutions under certain conditions, thus addressing a specific open problem.

Contribution

It establishes necessary and sufficient conditions for finite order solutions with exponential sum coefficients and describes their explicit form, advancing understanding of such differential equations.

Findings

Finite order solutions exist under specific conditions on coefficients.

Explicit form of solutions as exponential sums is provided.

Addresses a partial open problem in the theory of differential equations.

Abstract

We show a necessary and sufficient condition on the existence of finite order entire solutions of linear differential equations $$ f^{(n)}+a_{n-1}f^{(n-1)}+\cdots+a_1f'+a_0f=0,\eqno(+) $$ where $a_i$ are exponential sums for $i=0,\ldots,n-1$ with all positive (or all negative) rational frequencies and constant coefficients. Moreover, under the condition that there exists a finite order solution of (+) with exponential sum coefficients having rational frequencies and constant coefficients, we give the precise form of all finite order solutions, which are exponential sums. It is a partial answer to Gol'dberg-Ostrovski\v{i} Problem and Problem 5 in \cite{HITW2022} since exponential sums are of completely regular growth.