We derive evolution equations for the truncated Mellin moments of the parton distributions. We find that the equations have the same form as those for the partons themselves. The modified splitting function for n-th moment P ′ (n, x) is x n P (x), where P (x) is the well-known splitting function from the DGLAP equation. The obtained equations are exact for each n-th moment and for every truncation point x0 ∈ (0; 1). They can be solved with use of standard methods of solving the DGLAP equations.This approach allows us to avoid the problem of dealing with the unphysical region x → 0. Furthermore, it refers directly to the physical values -moments (rather than to the parton distributions), what enables one to use a wide range of deep-inelastic scattering data in terms of smaller number of parameters.We give an example of an application.

We review our previous studies of truncated Mellin moments of parton distributions. We show in detail the derivation of the evolution equation for double truncated moments. The obtained splitting function has the same rescaled form as in a case of the single truncated moments. We apply the truncated moments formalism to QCD analyses of the spin structure functions of the nucleon, g 1 and g 2 . We generalize the Wandzura-Wilczek relation in terms of the truncated moments and find new sum rules. We derive the DGLAP-like evolution equation for the twist-2 part of g 2 and solve it numerically. We find also useful relations between the truncated and untruncated moments.

We review the main results on the generalization of the DGLAP evolution equations within the cut Mellin moments (CMM) approach, which allows one to overcome the problem of kinematic constraints in Bjorken x. CMM obtained by multiple integrations as well as multiple differentiations of the original parton distribution also satisfy the DGLAP equations with the simply transformed evolution kernel. The CMM approach provides novel tools to test QCD; here we present one of them. Using appropriate classes of CMM, we construct the generalized Bjorken sum rule that allows us to determine the Bjorken sum rule value from the experimental data in a restricted kinematic range of x. We apply our analysis to COMPASS data on the spin structure function g1.