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Oscillatory properties of the second order half-linear difference equation $$\Delta(r_k|\Delta y_k|^{\alpha-2}\Delta y_k)+p_k|y_{k+1}|^{\alpha-2}y_{k+1}=0,$$ where $\alpha>1$, are investigated. A particular attention is devoted to the connection with oscillation theory of its continuous counterpart, half--linear differential equation and also with the theory of linear differential and difference equations. We present not only the overview of the existing results but we also establish new oscillation and nonoscillation criteria.