Power of a meromorphic function sharing value with its k-th order directional derivative in C^m

TL;DR

This paper extends uniqueness theorems for meromorphic functions sharing a value with their derivatives from one complex variable to multiple variables, providing new insights and optimal examples in higher dimensions.

Contribution

It generalizes previous results to several complex variables and demonstrates the sharpness of these theorems with illustrative examples.

Findings

Extended uniqueness theorems to $ ext{C}^m$

Provided examples showing optimality of results

Enhanced understanding of value sharing in higher dimensions

Abstract

In the context of several complex variables, we investigate the uniqueness problem for a power of a meromorphic function that shares a value with its $k$-th order directional derivative in $\mathbb{C}^m$. Our results extend previous uniqueness theorems from the one-variable case to higher dimensions. Furthermore, we provide numerous examples to demonstrate that our results are, in certain senses, best possible.