  In this contribution, we make use of the QCD perturbative fragmentation
function formalism to describe the one-particle inclusive fragmentation of a
heavy quark produced in $e^+e^-$ annihilation at $\mathcal{O}(\alpha_S^2)$. We
perform the computation analytically in Mellin-space. We resum soft-gluons
effects in initial conditions and coefficient functions and perform evolution
up to NNLL accuracy, obtaining the first NNLO + NNLL prediction for charm quark
production. We study the impact of different matching schemes and Landau pole
prescriptions in soft-gluon resummation, finding significant differences. We
extract simple non-perturbative fragmentation functions for $B$ and $D^*$
mesons by comparing the perturbative prediction with the data from CLEO, BELLE
and LEP experiments. We find that for charm mesons the experimental results
from CLEO/BELLE and from LEP are not reconcilable with the standard DGLAP
evolution.
