We extract the small x asymptotic behaviour of the Altarelli-Parisi splitti
ng 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 th
e choice of factorization scheme, and its all-order generalization. We deri
ve necessary conditions which must be fulfilled in order to obtain a well b
ehaved perturbative expansion, and show that they may be satisfied by a sui
table reorganization of the original series. (C) 1999 Published by Elsevier
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