Uniqueness theorems of meromorphic functions with their differential-difference operators in several complex variables

TL;DR

This paper investigates the uniqueness of meromorphic functions sharing small functions with their shifts and derivatives, extending previous results from entire to meromorphic functions and from derivatives to differential-difference polynomials.

Contribution

It generalizes existing uniqueness theorems to meromorphic functions involving differential-difference operators and small functions, improving upon prior entire function results.

Findings

Established conditions under which meromorphic functions are equal to their shifts.

Derived criteria for functions sharing small functions to be periodic or identical.

Extended the scope of uniqueness theorems to include meromorphic functions with finite order less than one.

Abstract

An example in the article shows that the first derivative of $f(z)=\frac{2}{1-e^{-2z}}$ sharing $0$ CM and $1,\infty$ IM with its shift $\pi i$ cannot obtain they are equal. In this paper, we study the uniqueness of meromorphic function sharing small functions with their shifts concerning its $k-th$ derivatives. We improves the author's result \cite{h} from entire function to meromorphic function, the first derivative to its differential-difference polynomial, and also finite values to small functions. As for $k=0$, we obtain: Let $f(z)$ be a transcendental meromorphic function of $\rho_{2}(f)<1$, let $c$ be a nonzero finite value, and let $a_{1}(z)\not\equiv\infty, a_{2}(z)\not\equiv\infty\in \hat{S}(f)$ be two distinct small functions of $f(z)$ such that $a(z)$ is a periodic function with period $c$ and $b(z)$ is any small function of $f(z)$. If $f(z)$ and $f(z+c)$ share $a_{1}(z),\infty$ CM, and share $a_{2}(z)$ IM, then either $f(z)\equiv f(z+c)$ or $$e^{p(z)}\equiv \frac{f(z+c)-a_{1}(z+c)}{f(z)-a_{1}(z)}\equiv \frac{a_{2}(z+c)-a_{1}(z+c)}{a_{2}(z)-a_{1}(z)},$$ where $p(z)$ is a non-constant entire function of $\rho(p)<1$ such that $e^{p(z+c)}\equiv e^{p(z)}$.