The modified evolution equation for parton distributions of Dokshitzer, Marchesini and Salam is extended to non-singlet Deep Inelastic Scattering coefficient functions and the physical evolution kernels which govern their scaling violation. Considering the x->1 limit, it is found that the leading next-to-eikonal logarithmic contributions to the physical kernels at any loop order can be expressed in term of the one-loop cusp anomalous dimension, a result which can presumably be extended to all orders in (1-x), and has eluded so far threshold resummation. Similar results are shown to hold for fragmentation functions in semi-inclusive e+ e- annihilation. Gribov-Lipatov relation is found to be satisfied by the leading logarithmic part of the modified physical evolution kernels.Comment: 12 pages; version 2: eq.(4.6) and comment below corrected (main results unchanged), section on fragmentation functions added; version 3: new results added: all-order relation in section 3.5, O(1-x) terms adressed in section 5; version 4: incorrect suggestion on Gribov-Lipatov reciprocity removed; version 5: slight extension of the version to be published in Physics Letters B, contains a discussion of O((1-x)^2) terms and added reference
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