Factorial Series Representation of Stieltjes Series Converging Factors

TL;DR

This paper introduces a method to express Stieltjes series converging factors as inverse factorial series, facilitating their analysis and potentially advancing the theoretical understanding of Weniger's transformation.

Contribution

It presents a recursive algorithm to analytically retrieve inverse factorial series for Stieltjes series converging factors, aiding convergence analysis.

Findings

Algorithm effectively retrieves inverse factorial series.

Applications demonstrate ease of implementation.

Potential for developing a convergence theory for Weniger's transformation.

Abstract

The practical usefulness of Levin-type nonlinear sequence transformations as numerical tools for the summation of divergent series or for the convergence acceleration of slowly converging series, is nowadays beyond dispute. Weniger's transformation, in particular, is able to accomplish spectacular results when used to overcome resummation problems, often outperforming better known resummation techniques, the most known being Pad\'e approximants. However, our understanding of its theoretical features is still far from being satisfactory and particularly bad as far as the decoding of factorially divergent series is concerned. Stieltjes series represent a class of power series of fundamental interest in mathematical physics. In the present paper, it is shown how the Stieltjes series converging factor of any order is expressible as an inverse factorial series, whose terms can be analytically retrieved through a simple recursive algorithm. A few examples of applications of our algorithm are presented, in order to show its effectiveness and implementation ease. We believe the results presented here could constitute an important, preliminary step for the development of a general convergence theory of Weniger's transformation on Stieltjes series. A rather ambitious project, but worthy of being pursued in the future.