Asymptotic Convergence of Weniger's $\delta$-Transformation for a Class of Superfactorially Divergent Stieltjes Series

TL;DR

This paper demonstrates that Weniger's δ-transformation effectively resums superfactorially divergent Stieltjes series, with proven asymptotic convergence rates and strong numerical support, offering a robust alternative to Padé approximants.

Contribution

It provides a rigorous asymptotic analysis of Weniger's δ-transformation applied to a specific class of superfactorially divergent series, establishing its convergence properties.

Findings

Weniger's δ-transformation converges asymptotically for the series studied.

An exact integral representation for the truncation error is derived.

Numerical experiments confirm the theoretical convergence rates.

Abstract

The resummation of superfactorially divergent series represents a significant computational challenge in mathematical physics. In the present paper the resummation of a specific class of Stieltjes series characterized by a moment sequence growing as $(2n)!$ will be addressed. Despite the fact that Carleman's condition is satisfied for these series, the convergence rate of Pad\'e approximants is severely hindered by the logarithmic divergence of the associated Carleman series. Weniger's $\delta$ transformation is proposed as a highly efficient alternative resummation tool. By employing recently established results on the converging factors of superfactorially divergent Stieltjes series, an exact integral representation for the truncation error is obtained. This representation enables the rigorous derivation of the leading-order asymptotic behavior of the transformation error, as well as the estimation of the related convergence rate, for real positive arguments. Numerical experiments strongly support the theoretical findings, suggesting that the $\delta$ transformation offers a robust and computationally efficient framework for decoding this class of wildly divergent expansions