Perturbative heavy quark fragmentation function through O(αs2): Gluon initiated contribution

We derive the gluon initiated contribution to the initial condition for the perturbative fragmentation function of a heavy quark through order O(αs2) in the MS scheme. This result is needed for the resummation with next-to-next-to-leading logarithmic accuracy of quasicollinear logarithms lnk(m2) in heavy quark differential distributions by solving the complete Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation. Together with the previously evaluated fermion initiated components, this result completes the derivation of the initial condition for the perturbative fragmentation function at next-to-next-to-leading order