Borel Transform and Scale-Invariant Fractional Derivatives United

The method of Borel transformation for the summation asymptotic expansions with power-law behavior at infinity is combined elements scale-invariant fractional analysis goal calculating critical amplitudes. order specially designed derivatives u used as a control parameter to be defined uniquely from u-optimization. For resummation transformed expansions, we employed self-similar iterated roots. We also consider complementary optimization, called b-optimization number iterations b an alternative parameter. Fractional Summation consists three constructive steps. first step corresponds u-optimization amplitudes fixed b. When fails, second u. However, when two steps fail, third simplified, Borel-light technique. marginal amplitude should found by means roots constructed series, optimized either above approaches and corrected diagonal Padé approximants. examples are given optimizations,“horses-for-courses” approach outperforms other analytical methods in calculation