[Paper Review] Summation of Divergent Power Series by Means of Factorial Series

Ernst Joachim Weniger|arXiv (Cornell University)|May 4, 2010

Scientific Measurement and Uncertainty Evaluation42 references4 citations

TL;DR

This paper demonstrates that factorial series—historically underappreciated tools—provide a powerful numerical method for summing divergent inverse power series, such as the asymptotic expansion of the exponential integral $E_1(z)$ and the perturbation series for the quartic anharmonic oscillator. By leveraging Stirling numbers and orthogonal/triangular matrix transformations, the method converts divergent series into convergent factorial series, enabling accurate numerical evaluation where standard summation fails.

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.

Motivation & Objective

• To demonstrate that factorial series are underutilized but highly effective tools for summing divergent (inverse) power series.
• To address the numerical challenge of summing asymptotic expansions that diverge beyond a finite number of terms.
• To establish a formal framework linking inverse power series and factorial series via Stirling numbers and generalized linear transformations.
• To apply the method to concrete physical problems, including the exponential integral and the quartic anharmonic oscillator, to validate its practical utility.

Proposed method

• The paper uses Stirling numbers of the first and second kind to derive transformation formulas between inverse power series and factorial series.
• It formulates the transformation as a matrix operation involving lower-triangular and orthogonal matrices, generalizing finite and infinite series expansions.
• The method converts divergent asymptotic series into convergent factorial series of the form $\Omega(z) = \sum_{\nu=0}^{\infty} \frac{a_\nu \nu!}{(z)_{\nu+1}}$, where $(z)_{\nu+1}$ is the Pochhammer symbol.
• The transformation is applied to the divergent asymptotic expansion of the exponential integral $E_1(z)$, yielding a convergent factorial series representation.
• The method is extended to the Rayleigh-Schrödinger perturbation series for the quartic anharmonic oscillator, transforming the divergent power series into a convergent factorial series.
• Theoretical justification is provided through generalized linear transformations, showing that Stirling number identities are special cases of broader orthogonal and triangular matrix relationships.

Experimental results

Research questions

• RQ1 Can factorial series effectively sum divergent inverse power series that arise in asymptotic expansions of special functions?
• RQ2 How do Stirling numbers facilitate the transformation between inverse power series and factorial series?
• RQ3 To what extent can the method be generalized beyond Stirling number identities to other orthogonal and triangular matrix transformations?
• RQ4 Can the method produce accurate numerical results for physically relevant divergent series, such as those in quantum mechanics?
• RQ5 What is the role of Pochhammer symbols and factorial coefficients in stabilizing the convergence of transformed series?

Key findings

• The transformation of the divergent asymptotic expansion of the exponential integral $E_1(z)$ into a factorial series results in a convergent representation that enables accurate numerical evaluation.
• The method successfully sums the factorially divergent Rayleigh-Schrödinger perturbation series for the ground state energy of the quartic anharmonic oscillator by converting it into a convergent factorial series.
• Stirling numbers of the first and second kind are shown to be central to the transformation between inverse power series and factorial series, forming the core of the mathematical machinery.
• The relationships between inverse powers and inverse Pochhammer symbols via Stirling numbers are identified as special cases of more general orthogonal and triangular matrix transformations.
• The paper establishes that the transformation formulas are formally valid and can be extended to infinite series expansions, provided convergence is verified case by case.
• The study reveals that factorial series are underappreciated in modern mathematical and physical literature, despite their proven numerical utility and theoretical foundation.