Self-similar continued root approximants

TL;DR

This paper introduces a new self-similar root approximant method for summing asymptotic series, which are common in physics, providing a convergent approach that generalizes continued fractions and Padé approximants.

Contribution

The paper develops a novel self-similar continued root method for extrapolating asymptotic series, with proven convergence and broad applicability.

Findings

Method generalizes continued fractions and Padé approximants.

Convergence of the method is rigorously proved.

Illustrated with examples from condensed-matter physics.

Abstract

A novel method of summing asymptotic series is advanced. Such series repeatedly arise when employing perturbation theory in powers of a small parameter for complicated problems of condensed matter physics, statistical physics, and various applied problems. The method is based on the self-similar approximation theory involving self-similar root approximants. The constructed self-similar continued roots extrapolate asymptotic series to finite values of the expansion parameter. The self-similar continued roots contain, as a particular case, continued fractions and Pad\'{e} approximants. A theorem on the convergence of the self-similar continued roots is proved. The method is illustrated by several examples from condensed-matter physics.