Summation of Divergent Power Series by Means of Factorial Series

TL;DR

This paper demonstrates that factorial series are effective numerical tools for summing divergent power series, including asymptotic expansions and perturbation series, by leveraging Stirling numbers and related transformations.

Contribution

It revisits factorial series, illustrating their utility in summing divergent series and connecting Stirling numbers to broader orthogonal transformations.

Findings

Successfully summed divergent asymptotic expansion of exponential integral

Applied factorial series to divergent Rayleigh-Schrödinger perturbation series

Highlighted the role of Stirling numbers in these transformations

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-Schr\"{o}dinger 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.