Converging Factors of a Class of Superfactorially Divergent Stieltjes Series

Padé approximants are computational tools customarily employed for resumming divergent Stieltjes series. However, they become ineffective or even fail when applied to Stieltjes series whose moments do not satisfy the Carleman condition. Differently from Padé, Levin-type transformations incorporate important structural information on the converging factors of a typical Stieltjes series. For example, the computational superiority of Weniger’s $\delta$-transformation over Wynn’s epsilon algorithm is ultimately based on the fact that Stieltjes series converging factors can always be represented as inverse factorial series. In the present paper, the converging factors of an important class of superfactorially divergent Stieltjes series are investigated via an algorithm developed one year ago from the first-order difference equation satisfied by the Stieltjes series converging factors. Our analysis includes the analytical derivation of the inverse factorial representation of the moment ratio sequence of the series under investigation, and demonstrates the numerical effectiveness of our algorithm, together with its implementation ease. Moreover, a new perspective on the converging factor representation problem is also proposed by reducing the recurrence relation to a linear Cauchy problem whose explicit solution is provided via Faà di Bruno’s formula and Bell’s polynomials.