Get the general expression of $\mathbb{E}(Z_t \epsilon_{t-k})$ for $k=0,1,2,\cdots$ in the $AR(p)$ models,
$$ Z_t = \phi_1 Z_{t-1} + \phi_2 Z_{t-2} + \cdots + \phi_p Z_{t-p} + \epsilon_t, \forall t=\cdots, 0, 1,2,\cdots $$
where $Z_t$ is the random variable, and $\epsilon_t \overset{iid}{\sim} Normal(0,\sigma^2)$.
$\ast$ This problem assumes the knowledge of autoregressive models, but I would like to ask here rather than Cross Validated community due to the problem's mathematical nature.
Try
$$ \begin{aligned} \mathbb{E}(Z_t \epsilon_t) &= \sigma^2 \\[7pt] \mathbb{E}(Z_t \epsilon_{t-1}) &= \phi_1\sigma^2 \\[7pt] \mathbb{E}(Z_t \epsilon_{t-2}) &= \phi_1^2\sigma^2 + \phi_2\sigma^2 = (\phi_1^2 + \phi_2)\sigma^2 \\[7pt] \mathbb{E}(Z_t \epsilon_{t-3}) &= \phi_1(\phi_1^2 + \phi_2)\sigma^2 + \phi_2(\phi_1\sigma^2) + \phi_3\sigma^2 = (\phi_1^3 + 2\phi_1\phi_2 + \phi_3)\sigma^2 \\[7pt] \mathbb{E}(Z_t \epsilon_{t-4}) &= \phi_1(\phi_1^3 + 2\phi_1\phi_2+\phi_3)\sigma^2 + \phi_2(\phi_1^2 + \phi_2)\sigma^2 + \phi_1\phi_3\sigma^2 + \phi_4\sigma^2 = (\phi_1^4 + 3\phi_1^2\phi_2 + +2\phi_1\phi_3 + \phi_2^2 + \phi_4)\sigma^2 \\[7pt] \end{aligned} $$
I do not see any patterns here, and it is expected that from
$$ \mathbb{E}(Z_t \epsilon_{t-p}) $$
some different pattern may arise.
Can someone formulate this problem in the context of Linear difference equation?
Any help will be appreciated.
You have an $AR(p)$ process $$ (1-\phi_1 B - \phi_2 B^2 - \dots - \phi_p B^p)Z = \varepsilon. $$ where $B\colon\mathbb{R^Z}\to\mathbb{R^Z}$, $(Z_t)\mapsto(Z_{t-1})$ is the backshift operator (aka lag operator). So assuming $Z$ is stationary, you can invert to get an $MA(\infty)$ representation $$ Z=(1-\phi_1 B - \phi_2 B^2 - \dots - \phi_p B^p)^{-1}\varepsilon. $$ So your task is, up to the factor $\sigma^2$, to find the coefficients of $X^k$ in the series expansion of $$ (1-\phi_1 X - \phi_2 X^2 - \dots - \phi_p X^p)^{-1}. $$