Renormalon-based resummation for spacelike and timelike QCD quantities whose perturbation expansion has general form

We present a generalisation of our previous approach of a renormalon-motivated resummation of the QCD observables. Previously it was applied to the spacelike observables whose perturbation expansion was $D(Q^2)=a(Q^2)+O(a^2)$, where $a(Q^2)\equiv {\alpha }_s(Q^2)\mathrm{/}\pi$ is the running QCD coupling. Now we generalise the resummation to spacelike quantities $D(Q^2)=a(Q^2{)}^{{\nu }_0}+O(a^{{\nu }_0+1})$ and timelike quantities $F(\sigma )=a(\sigma {)}^{{\nu }_0}+O(a^{{\nu }_0+1})$, where ${\nu }_0$ is in general a noninteger number ($0<{\nu }_0\le 1$). We evaluate with this approach a timelike quantity, namely the scheme-invariant factor of the Wilson coefficient of the chromomagnetic operator in the heavy-quark effective Lagrangian, and related quantities.