Meromorphic Functions Sharing One Value with Their Derivatives Concerning the Difference Operator

Abstract

Let f(z) be a non-constant meromorphic function of finite order, $$c\in \mathbb {C}\setminus \{0\}$$ and $$k\in \mathbb {N}$$ . Suppose f(z) and $$f^{(k)}(z+c)$$ share 1 CM (IM), f(z) and $$f(z+c)$$ share $$\infty $$ CM. If $$N(r,0;f)=S(r,f) \left( N\left( r,0;f(z)\right) +N\left( r,0;f^{(k)}(z+c)\right) =S(r,f)\right) $$ , then either $$f(z)\equiv f^{(k)}(z+c)$$ or f(z) is a solution of the following equation: $$\begin{aligned}f\; \hbox {and}\\&\quad N\left( r,0;f(z)+\frac{1}{a(z)}\right) =S(r,f)\\&\quad \left( f'(z+c)-1=a(z)\left( f(z)-1\right) \left( f(z)+\frac{1}{a(z)}\right) \right) \end{aligned}$$ where $$a(z)\left( \not \equiv -\,1,0,\infty \right) \left( a(z)\left( \not \equiv 0,\infty \right) \right) $$ is a meromorphic function satisfying $$T(r,a)=S(r,f)$$ . Also we exhibit some examples to show that the conditions of our results are the best possible.