2024/06/01 by Riccardo Borghi, Borghi, Riccardo
Mathematics · #Computational Physics (physics.comp-ph) #Differential Equations and Numerical Methods #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Meromorphic and Entire Functions #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2407.03344
openalex publication_date 2024/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.