Fixed-target experiments permit the study of hadron production in the target fragmentation region. It is expected that the tagging of specific particles in the target fragments can be employed to introduce a bias in the hard scattering process towards a specific flavour content. The case of hadrons containing a heavy quark is particularly attractive because of the clear experimental signatures and the applicability of perturbative QCD. The standard approach to one-particle inclusive processes

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... sed on fragmentation functions is valid in the current fragmentation region and for large transverse momenta p_T in the target fragmentation region, but it fails for particle production at small p_T in the target fragmentation region. A collinear singularity, which cannot be absorbed in the standard way into the phenomenological distribution functions, prohibits the application of this procedure. This situation is remedied by the introduction of a new set of distribution functions, the target fragmentation functions. They describe particle production in the target fragmentation region, and can be viewed as correlated distribution functions in the momentum fractions of the observed particle and of the parton initiating the hard scattering process. It is shown in a next-to-leading-order calculation for the case of deeply inelastic lepton-nucleon scattering that the additional singularity can be consistently absorbed into the renormalized target fragmentation functions on the one-loop level. The formalism is derived in detail and is applied to the production of heavy quarks. The renormalization group equation of the target fragmentation functions for the perturbative contribution is solved numerically, and the results of a case study for deeply inelastic lepton-nucleon scattering at DESY (H1 and ZEUS at HERA), at CERN (NA47) and at Fermilab (E665) are discussed. We also comment briefly on the case of an intrinsic heavy-quark content of the proton.