  We analyze two sets of specific functions, that/which form the basis of the
nonpower asymptotic expansions both in the timelike and spacelike regions for
single scale dependent QCD observables in the Shirkov--Solovtsov's Analytic
Perturbation Theory (APT) free of unphysical singularities. These functions are
explicitly derived up to the third order in the closed form in terms of the
Lambert-W function. As an input we used the exact two loop and the three loop
(corresponding to Pad\'e transformed beta-function) RG solutions for common
invariant coupling \alpha_s. The elegant recurrence formulas, helpful for
numerical analysis, are obtained for the both sets of the APT functions. Then
we construct the global versions of APT functions using the continuity
conditions (at the quark thresholds) on the \alpha_s in the \bar{MS} scheme and
give numerical results. For first three of these functions \mathfrak{A}_n(s)
and {\cal A}_n(Q^2); n=1,2,3 in the large interval of the momentum transfer and
energy (1 GeV <Q,{\sqrt s}< 170 GeV), numerical tables are presented. From
these we observe that, for the timelike arguments, the differences between
functions \mathfrak{A}_n(s) and the corresponding powers of the standard
iteratively approximated coupling \alpha_s^n(s) are not negligible even for
moderate energies in the five--flavor region.
