Value distribution and shared sets of differences of meromorphic functions

We investigate value distribution and uniqueness problems of difference polynomials of meromorphic functions. In particular, we show that for a finite order transcendental meromorphic function f with lambda(1/f) < rho(f) and a non-zero complex constant c, if n >= 2, then f (z)(n) f (z + c) assumes every non-zero value a is an element of C infinitely often. This research also shows that there exist two sets S(1) with 9 (resp. 5) elements and S(2) with 1 element, such that for a finite order nonconstant meromorphic (resp. entire) function f and a non-zero complex constant c, E(f(z)) (S(j)) = E(f(z+c))(S(j)) (j = 1.2) imply f (z) equivalent to f (z + c). This gives an answer to a question of Cross concerning a finite order meromorphic function f and its shift. (C) 2010 Elsevier Inc. All rights reserved.