Value distribution of exponential polynomials and their role in the theories of complex differential equations and oscillation theory

TL;DR

This paper reviews the history and current state of research on exponential polynomials' value distribution, highlighting their significance in complex differential equations and oscillation theory, and presents open problems for future exploration.

Contribution

It provides a comprehensive overview of exponential polynomial value distribution and discusses their applications in differential equations and oscillation theory, including new open problems.

Findings

Historical development from 1920 to 2021

Role of exponential polynomials in differential equations

Presentation of 13 open research problems

Abstract

An exponential polynomial is a finite linear sum of terms $P(z)e^{Q(z)}$, where $P(z)$ and $Q(z)$ are polynomials. The early results on the value distribution of exponential polynomials can be traced back to Georg P\'olya's paper published in 1920, while the latest results have come out in 2021. Despite of over a century of research work, many intriguing problems on value distribution of exponential polynomials still remain unsolved. The role of exponential polynomials and their quotients in the theories of linear/non-linear differential equations, oscillation theory and differential-difference equations will also be discussed. Thirteen open problems are given to motivate the readers for further research in these topics.