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.

Accepted for publication in Applied Numerical Mathematics, 15 pages, LaTeX2e, 0 figures

References in corpus (2)

• From Useful Algorithms for Slowly Convergent Series to Physical Predictions Based on Divergent Perturbative Expansions
• Verallgemeinerte Summationsprozesse als numerische Hilfsmittel für quantenmechanische und quantenchemische Rechnungen

Cited by in corpus (2)

• Rapidly Convergent Summation Formulas involving Stirling Series
• Extension of Summation Formulas involving Stirling series