Fractional Analytic QCD beyond Leading Order

Fractional analytic QCD is constructed beyond leading order using the standard inverse logarithmic expansion. It is shown that, contrary to the usual QCD coupling constant, for which this expansion can be used only for large values of its argument, in the case of analytic QCD, the inverse logarithmic expansion is applicable for all values of the argument of the analytic coupling constant. We present four different views, two of which are based primarily on Polylogarithms and generalized Euler $\zeta$-functions, and the other two are based on dispersion integrals. The results obtained up to the 5th order of perturbation theory, have a compact form and do not contain complex special functions that were used to solve this problem earlier. As an example, we apply our results to study the polarized Bjorken sum rule, which is currently measured very accurately.