A problem of Heittokangas-Ishizaki-Tohge-Wen concerning a certain differential-difference equation

TL;DR

This paper classifies all finite order entire solutions to a specific differential-difference equation involving polynomials and exponential functions, resolving an open problem posed by Heittokangas et al. in 2023.

Contribution

It provides a complete characterization of solutions to a particular differential-difference equation, solving an open problem in the field.

Findings

All finite order entire solutions are explicitly described.

The solutions depend on the polynomial coefficients and exponential terms.

The problem posed by Heittokangas-Ishizaki-Tohge-Wen is fully resolved.

Abstract

All the finite order entire solutions of \begin{equation*} f^n(z)+q(z)e^{Q(z)}f^{(k)}(z+c)=P(z) \end{equation*} are given, where $ q(z) $, $ Q(z), P(z) $ are polynomials, $ k $ and $ n \geq 2 $ are integers, and $ c \in \mathbb{C} \setminus \{0\} $.This solves an open problem of Heittokangas-Ishizaki-Tohge-Wen [Bull. Lond. Math. Soc. 55, 1-77 (2023)].