  The Lambert-W explicit solutions to the QCD renormalization group (RG)
equation are considered up to fourth order in the ${\bar {MS}}$ scheme. We
compare, systematically, these solutions with the conventional asymptotical
(iterative) approximations and with the exact numerical solutions to the RG
equation in the domain with three quark flavours. Applications of these
solutions in analytic perturbation theory (APT) are discussed. Using these
(Lambert-W, asymptotical and exact numerical) solutions we reconstruct the
expansion functions for the non-power APT series in the space- and time-like
regions. These expansion functions are examined in the infrared region. It is
shown that the Lambert-W solutions provide the excellent accuracy.
