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We study oscillatory properties of the second order half-linear difference equation (HL) Δ(rk|Δ yk|α−2 Δ yk)−pk|yk+1|α−2yk+1 = 0, α > 1. It will be shown that the basic facts of oscillation theory for this equation are essentially the same as those for the linear equation Δ(rkΔ yk)−pkyk+1 = 0. We present here the Picone type identity, Reid Roundabout Theorem and Sturmian theory for equation (HL). Some oscillation criteria are also given.