Uniqueness of meromorphic functions concerning their differential-difference operators

Suppose that f ( z ) is a nonconstant meromorphic function of hyper order strictly less than 1, c is a nonzero finite constant, a , b are distinct finite constants, and n ≥ 1 , k ≥ 0 are all integers. In this paper, we firstly prove one uniqueness theorem when f ( z ) and ( Δ c n f ( z ) ) ( k ) share a , ∞ CM and share b IM with additional conditions and give some further discussions. We also prove another uniqueness theorem when f ′ ( z ) and f ( z + c ) share ∞ CM and share a , b IM. Our main theorems generalize and improve many recent known results.