On periodic decomposition of entire functions of several variables

Consider the difference equation on : , where and . In this paper, we assume that c (1), . . . , c (m) are pairwise linearly independent over , except for the case m = 2. Firstly, we establish a general representation of its entire solutions. Secondly, under the condition L a parts per thousand currency sign n, we give a necessary and sufficient conditions for entire solutions to have representations as a sum of c (k) -periodic entire functions. Here L is the maximum integer such that are pairwise linearly independent over for some k (1),k (2), . . . ,k (L) . Finally, we show that every entire solution has a representation as a sum of c (k) -periodic meromorphic functions.