Dual exponential polynomials and a problem of Ozawa

TL;DR

This paper investigates complex differential equations with exponential polynomial coefficients, revealing a duality property between solutions and coefficients, and improves upon previous results while proposing open problems.

Contribution

It introduces new results on the growth and duality of solutions and coefficients in differential equations with exponential polynomial coefficients.

Findings

Order of solution equals order of dominant coefficient

Solution and coefficient exhibit a duality property

Advances previous results in the field

Abstract

Complex linear differential equations with entire coefficients are studied in the situation where one of the coefficients is an exponential polynomial and dominates the growth of all the other coefficients. If such an equation has an exponential polynomial solution $f$, then the order of $f$ and of the dominant coefficient are equal, and the two functions possess a certain duality property. The results presented in this paper improve earlier results by some of the present authors, and the paper adjoins with two open problems.