### Oscillation and nonoscillation of second order neutral delay difference equations

Some new oscillation and nonoscillation criteria for the second order neutral delay difference equation $$\Delta \left({c}_{n}\Delta ({y}_{n}+{p}_{n}{y}_{n-k})\right)+{q}_{n}{y}_{n+1-m}^{\beta}=0,\phantom{\rule{1.0em}{0ex}}n\ge {n}_{0}$$ where $k$, $m$ are positive integers and $\beta $ is a ratio of odd positive integers are established, under the condition ${\sum}_{n={n}_{0}}^{\infty}\frac{1}{{c}_{n}}<\infty .$