Optimal Mittag–Leffler Summation

Abstract

A novel method of an optimal summation is developed that allows for calculating from small-variable asymptotic expansions the characteristic amplitudes for variables tending to infinity. The method is developed in two versions, as the self-similar Borel–Leroy or Mittag–Leffler summations. It is based on optimized self-similar iterated roots approximants applied to the Borel–Leroy and Mittag–Leffler- transformed series with the subsequent inverse transformations. As a result, simple and transparent expressions for the critical amplitudes are obtained in explicit form. The control parameters come into play from the Borel–Leroy and Mittag–Leffler transformations. They are determined from the optimization procedure, either from the minimal derivative or minimal difference conditions, imposed on the analytically expressed critical amplitudes. After diff-log transformation, virtually the same procedure can be applied to critical indices at infinity. The results are obtained for a number of various examples. The examples vary from a rapid growth of the coefficients to a fast decay, as well as intermediate cases. The methods give good estimates for the large-variable critical amplitudes and exponents. The Mittag–Leffler summation works uniformly well for a wider variety of examples.