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

Xuxu Xiang, Jianren Long

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)].

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 , are polynomials, and are integers, and .This solves an open problem of Heittokangas-Ishizaki-Tohge-Wen [Bull. Lond. Math. Soc. 55, 1-77 (2023)].
Paper Structure (2 sections, 5 theorems, 21 equations)

This paper contains 2 sections, 5 theorems, 21 equations.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem \ref{['th 1.4']}

Key Result

Theorem 1.1

hayman Let meromorphic functions $f$ and $g$ satisfy the nonlinear differential equation where $P_d(z,f)$ is a differential polynomial in $f$ of degree $d\leq{n-1}$ with small functions coefficients. If $N(r,f)+N(r,\frac{1}{g})=S(r,f)$, then $g(z)=(f(z)+a(z))^n$, where $a$ is a small meromorphic function of $f$.

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Example 1.1
  • Example 1.2
  • Corollary 1.5