  We are concerned with optimal linear estimation of means on subsequent
occasions under sample rotation where evolution of samples in time is designed
through a cascade pattern. It has been known since the seminal paper of
Patterson (1950) that when the units are not allowed to return to the sample
after leaving it for certain period (there are no gaps in the rotation
pattern), one step recursion for optimal estimator holds. However, in some
important real surveys, e.g. Current Population Survey in the US or Labour
Force Survey in many countries in Europe, units return to the sample after
being absent in the sample for several occasions (there are gaps in rotation
patterns). In such situations difficulty of the question of the form of the
recurrence for optimal estimator increases drastically. This issue has not been
resolved yet. Instead alternative sub-optimal approaches were developed, as
K-composite estimation (see e.g. Hansen et al. (1955)), AK-composite estimation
(see e.g. Gurney and Daly (1965) or time series approach (see e.g. Binder and
Hidiroglou (1988)).
  In the present paper we overcome this long-standing difficulty, that is, we
present analytical recursion formulas for the optimal linear estimator of the
mean for schemes with gaps in rotation patterns. It is achieved under some
technical conditions: ASSUMPTION I and ASSUMPTION II (numerical experiments
suggest that these assumptions might be universally satisfied). To attain the
goal we develop an algebraic operator approach which allows to reduce the
problem of recursion for the optimal linear estimator to two issues: (1)
localization of roots (possibly complex) of a polynomial Q_p defined in terms
of the rotation pattern (Q_p happens to be conveniently expressed through
Chebyshev polynomials of the first kind), (2) rank of a matrix S defined in
terms of the rotation pattern and the roots of the polynomial Q_p.
