  We have calculated first and second order corrections to several sum rules
measured in deep inelastic lepton-hadron scattering. These corrections, which
are due to heavy flavors only, are compared with the existing perturbation
series which is computed for massless quarks up to third order in the strong
coupling constant \alpha_s. A study of the perturbation series reveals that the
large logarithms of the type ln^i Q^2/m^2 dominate the perturbation series at
much larger values than those given by the usual matching conditions imposed on
the \alpha_s(\mu). Therefore these matching conditions cannot be used to
extrapolate the running coupling constant from small \mu to very large scales
like \mu=M_Z. An alternative description of the running coupling constant in
the MOM-scheme is proposed.
