Factorial Series Representation of Stieltjes Series Converging Factors

TL;DR

The paper addresses the challenge of understanding and exploiting Weniger's Levin-type transformation for factorially divergent Stieltjes series by deriving a constructive inverse factorial representation of the series converging factors.A first-order difference equation for converging factors is solved via an explicit inverse factorial expansion, resulting in a triangular recurrence that computes the expansion coefficients from the moment-ratio sequence $\mu_m/\mu_{m+1}$.The authors connect this factorial representation to Weniger's transformation and provide an exact finite-difference formula to recover the Stieltjes function when the factorial expansion truncates, as well as a practical algorithm for generating the coefficients $c_k(z)$.Through several classical and general examples (Euler series, error function, logarithm, Lerch transcendent), the work demonstrates the method's effectiveness, including closed-form expressions in important special cases and links to Laguerre and Jacobi polynomials.Overall, the study advances a potential path toward a systematic convergence theory for Weniger-type transformations on Stieltjes series and paves the way for broader applicability to factorially divergent perturbation problems.

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 Padé 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.