Abstract
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 $g_1$.
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