The oscillatory behavior of the generalized first order nonlinear delay α-difference equation is of the form u(k + ℓ) - αu(k) + p(k)uγ(k - nℓ) = 0, α > 0, k ∈ [0,∞) (1) where p(k) is nonnegative real valued function, n is a positive integer, and γ ∈ (0,∞) is a quotient of odd positive integers is investigated. A necessary and sufficient condition of oscillation for sublinear equation (1) (0 < γ < 1) and an almost sharp condition of oscillation for superlinear equation (1) (γ > 1) are obtained. AMS Subject Classification: 39A10, 39A11, 39A12. © 2016 Academic Publications, Ltd.