(a) Show that (left(1-mathrm{B}^{2} ight)left(x_{t} ight)=(1-mathrm{B})(1+mathrm{B})left(x_{t} ight)). (b) Show that (triangle^{k}left(x_{t} ight)=sum_{h=0}^{k}(-1)^{h}binom{k}{h} x_{t-h}) (equation (5.26)). (c) Show...

(a) Show that \(\left(1-\mathrm{B}^{2}\right)\left(x_{t}\right)=(1-\mathrm{B})(1+\mathrm{B})\left(x_{t}\right)\).

(b) Show that \(\triangle^{k}\left(x_{t}\right)=\sum_{h=0}^{k}(-1)^{h}\binom{k}{h} x_{t-h}\) (equation (5.26)).

(c) Show that for any integer \(k,\left(1-\mathrm{B}^{k}\right)\left(x_{t}\right)=\triangle_{k}\left(x_{t}\right)\).