Factorial Series Representation of Stieltjes Series Converging Factors

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. The Weniger transformation, in particular, is able to accomplish spectacular results when used to overcome resummation problems, often outperforming better-known resummation techniques, like, for instance, Padé approximants. However, our theoretical understanding of Levin-type transformations is still far from being satisfactory and is particularly bad as far as the decoding of factorially divergent series is concerned. The Stieltjes series represent a class of power series of fundamental interest in mathematical physics. In the present paper, it is shown how the converging factor of any order of typical Stieltjes series can be expressed as an inverse factorial series, whose terms are analytically retrieved through a simple recursive algorithm. A few examples of applications are presented, in order to show the effectiveness and implementation ease of the algorithm itself. We believe that further investigations of the asymptotic forms of the remainder terms, encoded within the converging factors, could eventually lead toward a more general theory of the asymptotic behavior of the Weniger transformation when it is applied to Stieltjes series in high transformation orders. It is a rather ambitious project, which should be worthy of being pursued in the future.