  We introduce stochastic sequences $\zeta(k)$ with periodically stationary
generalized multiple increments of fractional order which combines
cyclostationary, multi-seasonal, integrated and fractionally integrated
patterns. We solve the problem of optimal estimation of linear functionals
constructed from unobserved values of stochastic sequences $\zeta(k)$ based on
their observations at points $ k<0$. For sequences with known spectral
densities, we obtain formulas for calculating values of the mean square errors
and the spectral characteristics of the optimal estimates of functionals.
Formulas that determine the least favorable spectral densities and minimax
(robust) spectral characteristics of the optimal linear estimates of
functionals are proposed in the case where spectral densities of sequences are
not exactly known while some sets of admissible spectral densities are given.
