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 \(n\ge 1\) $n\ge 1$ , \(k\ge 0\) $k\ge 0$ are all integers. In this paper, we firstly prove one uniqueness theorem when f(z) and \((\Delta ^{n}_{c}f(z))^{(k)}\) ${({\mathrm{\Delta }}_c^{n}f(z))}^{(k)}$ share \(a,\infty\) $a,\infty$ CM and share b IM with additional conditions and give some further discussions. We also prove another uniqueness theorem when \(f'(z)\) $f^{\prime }(z)$ and \(f(z+c)\) $f(z+c)$ share \(\infty\) $\infty$ CM and share a, b IM. Our main theorems generalize and improve many recent known results.