Summation of divergent power series by means of factorial series

Abstract

Factorial series played a major role in Stirling's classic book Methodus Differentialis (1730), but now only a few specialists still use them. This article wants to show that this neglect is unjustified, and that factorial series are useful numerical tools for the summation of divergent (inverse) power series. This is documented by summing the divergent asymptotic expansion for the exponential integral E 1 (z) and the factorially divergent Rayleigh-Schrödinger perturbation expansion for the quartic anharmonic oscillator. Stirling numbers play a key role since they occur as coefficients in expansions of an inverse power in terms of inverse Pochhammer symbols and vice versa. It is shown that the relationships involving Stirling numbers are special cases of more general orthogonal and triangular transformations.

Numerical calculation of Bessel, Hankel and Airy functions

“…[24,43,44] (see also §3.9 of Ref. [13]), in many cases the following sequence transformation (Weniger transformation), (30) leads to an accelerated convergence (or summation in the case of divergence) of the input sequence {s n } ∞ n=0…”

Borel-Laplace summation method used as time integration scheme

“…Indeed, in some cases, it has to be combined to other techniques (such as singular value decomposition method [1,18]) to avoid the apparition of a singular matrix. As an alternative, factorial series may be used [15,27,28].…”

On the resummation of series of fuzzy numbers via generalized Dirichlet and generalized factorial series

“…On the other hand, factorial series are studied with regard to their analytical properties and used in transformation of series and in expansions of functions. Authors converted power series to factorial series of more rapid convergence and given factorial series expansions for some functions [7][8][9][10]. There are also studies concerning factorial series solutions of difference equations [11][12][13].…”

“…There are also studies concerning factorial series solutions of difference equations [11][12][13]. In connection with the context of the paper, both series are also utilized in regularization of divergent series [8,10,12,14,15].…”