Meromorphic functions that share values with their shifts or their n-th order differences

Abstract

Let f(z) be a non-constant meromorphic (entire) function of hyper-order strictly less than 1, n ≥ 3 (n ≥ 2) be an integer. It is shown that if fn(z) and fn(z + c) share a(≠ 0) ∈ ℂ and ∞ CM, then f(z) = t1f(z + c) or f(z) = t2f(z + 2c), where t1 and t2 satisfy $$t_i^n=1$$ , (i = 1, 2). Some examples are provided to show the sharpness of our results. In addition, we mainly obtain two uniqueness results of f(z) with their n-th order differences Δnf. For example, let f(z) be transcendental entire, and let a(≠ 0) ∈ ℂ. We show that, if f(z) and Δnf(≢ 0) share 0 CM and a IM, then f(z) = Δnf. And this research extends earlier results by Chen et al.