  We calculate the $O(\alpha_s^2)$ gluonic operator matrix elements for the
twist--2 operators, which contribute to the heavy flavor Wilson coefficients in
unpolarized deeply inelastic scattering in the region $Q^2 \gg m^2$, up to the
linear terms in the dimensional parameter $\varepsilon$, ($D= 4 +
\varepsilon$). These quantities are required for the description of parton
distribution functions in the variable flavor number scheme (VFNS). The
$O(\alpha_s^2 \varepsilon)$ terms contribute at the level of the
$O(\alpha_s^3)$ corrections through renormalization. We also comment on
additional terms, which have to be considered in the fixed (FFNV) and variable
flavor number scheme, adopting the $\overline{\rm MS}$ scheme for the running
coupling constant.
