[en] We present new sets of fragmentation functions for charged pions and kaons, both at leading and next-to-leading order. They are fitted to data on inclusive charged-hadron production in e⁺e^- annihilation taken by TPC at PEP (√s=29 GeV) and to similar data by ALEPH at LEP, who discriminated between events with charm, bottom, and light-flavour fragmentation in their charged-hadron sample. In contrast to our previous analysis, where we only distinguished between valence-quark, sea-quark, and gluon fragmentation, we are now able to treat all partons independently and to properly incorporate the charm and bottom thresholds. Due to the sizeable energy gap between PEP and LEP, we are sensitive to the scaling violation in the fragmentation process, which allows us to extract a value for the asymptotic scale parameter of QCD, Λ. Recent data on inclusive charged-hadron production in tagged three-jet events by OPAL and similar data for longitudinal electron polarization by ALEPH allow us to pin down the gluon fragmentation functions. Our new fragmentation functions lead to an excellent description of a multitude of other e⁺e^- data on inclusive charged-hadron production, ranging from √s=5.2 GeV to LEP energy. In addition, they agree nicely with the transverse-momentum spectra of single charged hadrons measured by H1 and ZEUS in photoproduction at the ep collider HERA, which represents a nontrivial check of the factorization theorem of the QCD-improved parton model. In this comparison, we also find first evidence for the interplay between the direct-and resolved-photon mechanisms and for the existence of a gluon density inside the photon. (orig.)