The non-singlet QCD evolution kernels improved by renormalon chain   contributions
  Closed expressions are presented for the contributions to QCD non-singlet forward evolution kernels $P(z)$ for the DGLAP equation and to $V(x,y)$ for non-forward (ER-BL) evolution equation for a certain class of diagrams which include renormalon chains. Calculations are performed in covariant $\xi$-gauge, in the \MSbar scheme. The assumption of "naive nonabelianization" approximation for kernel calculations is discussed, and a special choice of the gauge parameter $\xi=-3$ is analyzed in this context. Partial solutions to the ER-BL evolution equation are obtained. 
S.V. Mikhailov