  We consider the effects connected with the detailed radiative transfer during
the epoch of cosmological recombination on the ionization history of our
Universe. We focus on the escape of photons from the hydrogen Lyman-alpha
resonance at redshifts 600<~ z <~ 2000, one of two key mechanisms defining the
rate of cosmological recombination. We approach this problem within the
standard formulation, and corrections due to two-photon interactions are
deferred to another paper. As a main result we show here that within a
non-stationary approach to the escape problem, the resulting correction in the
free electron fraction, N_e, is about ~1.6-1.8% in the redshift range
800<~z<~1200. Therefore the discussed process results in one of the largest
modifications to the ionization history close to the maximum of
Thomson-visibility function at z~1100 considered so far. We prove our results
both numerically and analytically, deriving the escape probability, and
considering both Lyman-alpha line emission and line absorption in a way
different from the Sobolev approximation. In particular, we give a detailed
derivation of the Sobolev escape probability during hydrogen recombination, and
explain the underlying assumptions. We then discuss the escape of photons for
the case of coherent scattering in the lab frame, solving this problem
analytically in the quasi-stationary approximation and also in the
time-dependent case. We show here that during hydrogen recombination the
Sobolev approximation for the escape probability is not valid at the level of
DP/P~5-10%. This is because during recombination the ionization degree changes
significantly over a characteristic time Dz/z~10%, so that at percent level
accuracy the photon distribution is not evolving (abridged)
