Generalizing the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi Evolution of Fragmentation Functions to the Smallest x Values

An approach valid to any order which unifies the fixed order Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution of fragmentation functions at large x with soft gluon logarithmic resummation at small x is proposed. At lowest order, this approach, implemented with the double logarithmic approximation, reproduces exactly the modified leading logarithm approximation but is more complete due to the degrees of freedom given to the quark sector and the inclusion of the fixed order terms. We find that data from the largest x values to the peak region can be better fitted than with other approaches