Resolving the Problem of Multiple Control Parameters in Optimized Borel-Type Summation

TL;DR

The paper tackles the persistent issue of multiple control-parameter solutions in optimized Borel-type summation of divergent series. It develops a cost-functional framework to select among competing control parameters, and introduces fractional-derivative and fractional-integral transformations within self-similar Borel-type summation, alongside lasso and ridge type selection criteria. Across numerous physical models, the proposed methods yield accurate large-variable amplitudes $B$ with known powers $\beta$, often outperforming traditional summation techniques. The work provides a practical pathway to stabilize resummation schemes and suggests directions for a global optimization approach to further enhance reliability.

Abstract

One of the most often used methods of summing divergent series in physics is the Borel-type summation with control parameters improving convergence, which are defined by some optimization conditions. The well known annoying problem in this procedure is the occurrence of multiple solutions for control parameters. We suggest a method for resolving this problem, based on the minimization of cost functional. Control parameters can be introduced by employing the Borel-Leroy or Mittag-Leffler transforms. Also, two novel transformations are proposed using fractional integrals and fractional derivatives. New cost functionals are advanced, based on lasso and ridge selection criteria, and their performance is studied for a number of models. The developed method is shown to provide good accuracy for the calculated quantities.