  In analogy with parton distribution functions, also parton fragmentation
functions obey matching conditions when crossing heavy-flavour thresholds. We
compute these matching conditions at next-to-leading order in the strong
coupling constant alpha_s in the MSbar scheme. Our results can be used for the
dynamical generation of the heavy-flavour component in next-to-leading order
fits to light-hadrons fragmentation functions. Furthermore, when computing
perturbatively the charm fragmentation function from first principles and
evolving it to higher scales, our matching conditions should be used for
consistency when crossing the bottom threshold.
