The next-to-next-to-leading order (NNLO) corrections in massless perturbative
QCD are derived for the parton distributions of the photon and the deep
inelastic structure functions F_1^gamma and F_2^gamma. We present the full
photonic coefficient functions at order alpha alpha_s and calculate the first
six even-integer moments of the corresponding O(alpha alpha_s^2) photon-quark
and photon-gluon splitting functions together with the moments of the alpha
alpha_s^2 coefficient functions which enter only beyond NNLO. These results are
employed to construct parametrizations of the splitting functions which prove
to be sufficiently accurate at least for momentum fractions x >= 0.05. We also
present explicit expressions for the transformation from the MS_bar to the
DIS_gamma factorization scheme and write down the solution of the evolution
equations. The numerical impact of the NNLO corrections is discussed in both
schemes.