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As you said, $B$ is the lag or backward shift operator such that $BX_t=X_{t-1}$ and $B^pX_t=X_{t-p}$. Let $A$ now be polynomial, say $A(x)=a_1 x + a_2x^2+...+a_px^p$. Then, \begin{align} A(B) X_t &= \left( a_1 B + a_2B^2+...+a_pB^p\right) X_t \\ &=a_1 X_{t-1} + a_2 X_{t-2} + ... + a_p X_{t-p} \end{align} and \begin{align} \big(1-A\big)(B) X_t &=...


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