Small x behaviour of Altarelli-Parisi splitting functions

We extract the small x asymptotic behaviour of the Altarelli-Parisi splitting functions from their expansion in leading logarithms of 1/x. We show in particular that the nominally next-to-leading correction extracted from the Fadin-Lipatov kernel is enhanced asymptotically by an extra ln1/x over the leading order. We discuss the origin of this problem, its dependence on the choice of factorization scheme, and its all-order generalization. We derive necessary conditions which must be fulfilled in order to obtain a well behaved perturbative expansion, and show that they may be satisfied by a suitable reorganization of the original series. (C) 1999 Published by Elsevier Science B.V. All rights reserved.