Meromorphic Solutions of Difference Equations Involving Borel and Nevanlinna Exceptional Values

TL;DR

This paper explores the explicit forms and existence of meromorphic solutions to certain difference equations involving functions with Borel or Nevanlinna exceptional values, filling a gap in the functional analysis of such equations.

Contribution

It characterizes the explicit general meromorphic solutions to specific difference equations involving sharing values and provides concrete examples validating the results.

Findings

Proved existence and characterized solutions to $L_{c}^{n}(f) \\equiv Af$.

Analyzed rational and transcendental solutions for second-order difference equations.

Provided examples demonstrating the necessity of conditions for solutions.

Abstract

The existence of meromorphic solutions to various difference equations has been extensively studied in recent years, the precise functional forms of such solutions -- particularly when the function and its difference operators share values -- remain largely unexplored. This paper addresses this research gap by investigating the sharing value problem between finite-order meromorphic functions $f(z)$ and their linear difference operators $L_{c}^{n}(f)$. Specifically, we consider functions having Borel or Nevanlinna exceptional values. We prove not only the existence but also characterize the explicit general meromorphic solutions to the difference equation $L_{c}^{n}(f)\equiv Af$ for $A\in\mathbb{C}\backslash\{0\}$. To validate our main results and demonstrate the necessity of our conditions, we provide several concrete examples. Furthermore, we investigate the existence and nature of both rational and transcendental meromorphic solutions for the second-order difference equation $b_{2}(z)f(z+2\eta)+b_{1}(z)f(z+\eta)+b_{0}(z)f(z)=b(z)$ with polynomial coefficients.