Unicity on entire function concerning its differential-difference operators

Abstract

In this paper, we study the uniqueness of the differential-difference polynomials of entire functions on ${\mathbb{C}}^n$. We prove the following result: Let $f(z)$ be a transcendental entire function on ${\mathbb{C}}^n$ of hyper-order less than $1$ and $g(z)=b_{-1}+{\sum }_{i=0}^n{b}_i{f}^{(k_i)}(z+{\eta }_i)$, where $b_{-1}$ and $b_i(i=0\dots ,n)$ are small meromorphic functions of $f$ on ${\mathbb{C}}^n$, $k_i\ge 0(i=0\dots ,n)$ are integers, and ${\eta }_i(i=0\dots ,n)$ are finite values. Let $a_1(z)\not\equiv \infty ,a_2(z)\not\equiv \infty$ be two distinct small meromorphic functions of $f(z)$ on ${\mathbb{C}}^n$. If $f(z)$ and $g(z)$ share $a_1(z)$ CM, and $a_2(z)$ IM. Then either $f(z)\equiv g(z)$ or $a_1=2a_2=2$, $$f(z)\equiv e^{2p}-2e^p+2,$$ and $$g(z)\equiv e^p,$$ where $p(z)$ is a non-constant entire function on ${\mathbb{C}}^n$. Especially, in the case of $g(z)=({\Delta }_{\eta }^{n}f(z){)}^k$, we obtain $f(z)\equiv ({\Delta }_{\eta }^{n}f(z){)}^k$.

15 pages, Comments Welcomed

References in corpus (1)

• Nevanlinna theory for the difference operator

Cited by in corpus (1)

• Some results in the uniqueness of meromorphic function