This paper is concerned with the oscillation of all solutions of the delay difference equations and where {p_n} and {p_i(n)} are sequences of nonnegative real numbers an k and k_i are positive integers. New oscillation criteria of the forms for equation (*) and for equation (**) are obtained, where α = lim inf_n→∞ (1/k) Σ^n−1i=n−k p_i and C(α) is the “the best possible” function of α in some sense.