Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Unicity on entire function concerning its differential-difference
  operators   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
$b_{i} (i=0\ldots,n)$ are small meromorphic functions of $f$ on
$\mathbb{C}^{n}$, $k_{i}\geq0 (i=0\ldots,n)$ are integers, and $\eta_{i}
(i=0\ldots,n)$ 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 $f(z)\equiv
(\Delta_{\eta}^{n}f(z))^{k}$.

16
The categories of this paper are: stat
23
The categories of this paper are: solv-int
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
12
The categories of this paper are: nucl-th
17
The categories of this paper are: q-fin
4
The categories of this paper are: cond-mat
19
The categories of this paper are: eess
28
The categories of this paper are: alg-geom
14
The categories of this paper are: math-ph
30
The categories of this paper are: q-alg
22
The categories of this paper are: chao-dyn
36
The categories of this paper are: cmp-lg
9
The categories of this paper are: nlin
32
The categories of this paper are: atom-ph
24
The categories of this paper are: dg-ga