We study oscillatory properties of the second order half-linear difference equation $$\Delta (r_k|\Delta y_k{|}^{\alpha -2}\Delta y_k)-p_k{\left|y_{k+1}\right|}^{\alpha -2}{y}_{k+1}=0,\alpha >1.\left(\mathrm{HL}\right)$$ It will be shown that the basic facts of oscillation theory for this equation are essentially the same as those for the linear equation $$\Delta \left(r_k\Delta y_k\right)-p_k{y}_{k+1}=0.$$ We present here the Picone type identity, Reid Roundabout Theorem and Sturmian theory for equation (HL). Some oscillation criteria are also given.