Stirling-Markov series transformations with increasingly rapid convergence rates
A series transformation based on an inverse factorial integral representation is given. The procedure yields a hierarchy of transformations. Transformations of higher rank convergence faster and apply to larger classes of series. The procedure can be considered as a specific formulation of Markov's transformation of series, and the rank 1 transformation includes typical examples of Markov's transformation. It is shown that the rank K transformation of any rational series converges exponentially fast with ratio \(\rho_ K=K^ K/(K+1)^{K+1}\). Since \(\rho_ K\) tends to zero as K increases, it follows that every rational series can be transformed to a series converging at an arbitrarily fast exponential rate. The generalization of the procedure to other integral transforms is discussed.
scientific article; zbMATH DE number 3944448