The Gluonic Operator Matrix Elements at O(\alpha_s^2) for DIS Heavy Flavor Production

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.