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Some new oscillation and nonoscillation criteria for the second order neutral delay difference equation $$\Delta \left(c_n\Delta (y_n+p_n{y}_{n-k})\right)+q_n{y}_{n+1-m}^{\beta }=0,n\ge n_0$$ where $k$, $m$ are positive integers and $\beta$ is a ratio of odd positive integers are established, under the condition ${\sum }_{n=n_0}^{\infty }\frac{1}{c_n}<\infty .$