Answer to Question #215839 in Differential Equations for RAJ

Question

 Find Z[n(n – 1) (n – 2)].

Answer 1

The Z-transform of following function will be

$Z[n(n-1)(n-2)]=Z[n(n^{2}-3n+2)]=Z[n^{3}-3n^{2}+2n]$

$=Z[n^{3}]-3Z[n^{2}]+2Z[n]$

$=$ We know that $Z$ transform of $n^{3}$ is given by

$Z[n^{3}]=\dfrac{z^{3}+4z^{2}+z}{(z-1)^{4}}$

$Z[n^{2}]=\dfrac{z^{2}+z}{(z-1)^{3}}$

$Z[n]=\dfrac{z}{(z-1)^{2}}$

Now putting these values in above equation , we get

$=$ $\dfrac{z^{3}+4z^{2}+z}{(z-1)^{4}}+\dfrac{z^{2}+z}{(z-1)^{3}}+\dfrac{z}{(z-1)^{2}}$