A constructive theory for the general solution of kth-order difference equation is given as in a forthcoming paper of the author. As complement of the analytical theory [George D. Birkhoff, General theory of linear difference equations, Transactions of the American Mathematical Society, volume 12, number 2, pages 243–284, 1911], this constructive approach permits us an explicit and nonrecurrent representation of the general solution, for any initial conditions, and any sequences of complex numbers . If k = 1, then the solution is straightforward.