Question $5$ For a LTI system, given the input sequence

$f[k]=(\frac{1}{3}{)}^{k}u[k],$

the output sequence is

$y[k]=\frac{5}{2}(\frac{1}{3}{)}^{k}u[k]-\frac{3}{2}(\frac{1}{5}{)}^{k}u[k].$

$(1)$ Determine the transfer function $H(z)$ of the LTI system.

$(2)$ Determine the impulse response $h[k]$ of the system.

$(3)$ Determine the difference$-$equation representation of the system.