Unicity of Meromorphic Functions Concerning Differential-Difference Polynomials

Abstract

In this paper, we study unicity of meromorphic functions concerning differential-difference polynomials and mainly prove: Let \(k_{1},k_{2},\cdots,k_{n}\) be nonnegative integers and \(k=\) max\(\{k_{1},k_{2},\cdots,k_{n}\}\), let \(l\) be the number of distinct values of \(\{0,c_{1},c_{2},\cdots,c_{n}\}\), let \(s\) be the number of distinct values of \(\{c_{1},c_{2},\cdots,c_{n}\}\), let \(f(z)\) be a nonconstant meromorphic function of finite order satisfying \(N(r,f)\leq\frac{1}{8(lk+l+2s-1)+1}T(r,f)\), let \(m_{1}(z),m_{2}(z),\cdots,m_{n}(z),\) \(a(z),b(z)\) be small functions of \(f(z)\) such that \(a(z)\not\equiv b(z)\), let \((c_{1},k_{1}),(c_{2},k_{2}),\) \(\cdots,(c_{n},k_{n})\) be distinct and let \(F(z)=m_{1}(z)f^{(k_{1})}(z+c_{1})+m_{2}(z)f^{(k_{2})}(z+c_{2})+\cdots+m_{n}(z)f^{(k_{n})}(z+c_{n})\). If \(f(z)\) and \(F(z)\) share \(a(z),b(z)\) CM, then \(f(z)\equiv F(z)\). Our results improve and extend some results due to [1, 18, 20].