Derivatives of Meromorphic Functions Sharing Polynomials with Their Difference Operators

Abstract

In this paper, we investigate the uniqueness of meromorphic functions of finite order \(f(z)\) concerning their difference operators \(\Delta_{c}f(z)\) and derivatives \(f^{\prime}(z)\) and prove that if \(\Delta_{c}f(z)\) and \(f^{\prime}(z)\) share \(a(z)\), \(b(z)\), \(\infty\) CM, where \(a(z)\) and \(b(z)\) are two distinct polynomials, then they assume one of following cases: \((1)\) \(f^{\prime}(z)\equiv\Delta_{c}f(z)\); \((2)\) \(f(z)\) reduces to a polynomial and \(f^{\prime}(z)-A\Delta_{c}f(z)\equiv(1-A)(c_{n}z^{n}+c_{n-1}z^{n-1}+\cdots+c_{1}z+c_{0})\), where \(A(\neq 1)\) is a nonzero constant and \(c_{n},c_{n-1},\cdots,c_{1},c_{0}\) are all constants. This generalizes the corresponding results due to Qi et al. and Deng et al.