SARIMA Notation

$$ SARIMA \left\{ (p,d,q) \times (P,D,Q)_m \right\} $$

Seasonal Difference

$$ \Delta_mY_t=Y_t-Y_{t-m} $$

SARIMA model

$$ Y_t=c+\sum_{n=1}^{p}\beta_nY_{t-n}+\sum_{n=1}^{q}\theta_n\varepsilon_{t-n}+\sum_{n=1}^{P}\phi_nY_{t-sn}+\sum_{n=1}^{Q}\eta_n\varepsilon_{t-sn}+\varepsilon_t $$

equation

SARIMAX model

$$ Y_t=c+\sum_{n=1}^{p}\beta_nY_{t-n}+\sum_{n=1}^{q}\theta_n\varepsilon_{t-n}+ \sum_{n=1}^r\beta_n x_{n_t} +\sum_{n=1}^{P}\phi_nY_{t-sn}+\sum_{n=1}^{Q}\eta_n\varepsilon_{t-sn}+\varepsilon_t $$

Reference

Time Series Forecasting with ARIMA , SARIMA and SARIMAX

https://www.youtube.com/watch?v=WjeGUs6mzXg