A time-series yi with no trend has seasonality of period c if E[yi] = E[yi+c].
If we have a stationary time series yi and a deterministic time series si such that si = si+c for all i (and so si = si+kc for all integers k), then zi = yi + si would be a seasonal time series with period c. As shown in Regression with Seasonality, the seasonality of such time series can be modeled by using c–1 dummy variables.
A second way to model seasonality is to assume that si = μm(i) + εi where εi is a purely random time series and μ0, …, μc-1 are constants where m(i) = MOD(i,c).
A third approach is to model seasonality as a sort of random walk, i.e. si = μm(i) + si-c + εi . If μ0 = … = μc-1 = 0 then there is no drift; otherwise μ0, …, μc-1 capture the seasonal drift.
Of course, seasonality can be modeled in many other ways.
Recall that for the lag function Lc(yi) = yi-c, and so (1–Lc)yi = yi – yi-c. This is the principal way of expressing seasonality for SARIMA models.
Note too that if si is deterministic, then (1–Lc)si = si – si-c= 0.
References
Greene, W. H. (2002) Econometric analysis. 5th Ed. Prentice-Hall
https://www.scirp.org/(S(351jmbntvnsjt1aadkposzje))/reference/referencespapers.aspx?referenceid=1243286
Gujarati, D. & Porter, D. (2009) Basic econometrics. 5th Ed. McGraw Hill
http://www.uop.edu.pk/ocontents/gujarati_book.pdf
Hamilton, J. D. (1994) Time series analysis. Princeton University Press
https://press.princeton.edu/books/hardcover/9780691042893/time-series-analysis
Wooldridge, J. M. (2009) Introductory econometrics, a modern approach. 5th Ed. South-Western, Cegage Learning
https://cbpbu.ac.in/userfiles/file/2020/STUDY_MAT/ECO/2.pdf