Solved – How to form ARIMA equation given MA and AR terms

From what I know of the ARIMA model (please correct me anyone if I'm wrong) is that it is equivivalent to "pre-filter" (differencing) the data $y(t)$ and set $y_{new}(t) = y(t)-y(t-1)$. Then fit an ARMA model to this new data. This would mean your ARMA model:

begin{equation} y_{new}(t) = (1-q^{-1}) y(t) = frac{C}{A}e(t), end{equation} where $q^{-1}$ is the difference operator $q^{-1}y(t) = y(t-1)$ and $e(t)$ is noise.

This means you get the model (in $y$) as begin{equation} (1 – q^{-1})(1+a_1q^{-1}+ldots+a_{n_a} q^{-n_a})y(t) = (1 + c_1 q^{-1} + ldots + c_{n_c} q^{-n_c})e(t). end{equation}