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

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 $n\ge 2$ 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)].