Higher order difference operators and uniqueness of meromorphic functions

It is a known uniqueness result that if f is a transcendental meromorphic function of finite order with two Borel excetional values $$a \ne \infty , b$$ and its first order difference operator $${\triangle }_c f\not \equiv 0$$ , for some complex number c, and if f and $${\triangle }_c f$$ share a, b CM, then $$a=0, b=\infty $$ and $$f=\exp (Az+B)$$ , where $$A\ne 0, B \in {\mathbb {C}} $$ . This type of results has its origin dating back to Csillag-Tumura’s uniqueness theorems. In this paper, by using completely different methods, we shall show that the result holds for arbitrary higher order difference operators. Examples are provided to show that this result is not valid for meromorphic functions with infinite order, which also shows a distinction between the derivatives and the difference operators of meromorphic functions, in view of Csillag-Tumura type uniqueness theorems.