From Useful Algorithms for Slowly Convergent Series to Physical Predictions Based on Divergent Perturbative Expansions

TL;DR

This review situates convergence acceleration and Borel resummation at the border of mathematics and theoretical physics, presenting a unified framework where nonlinear sequence transformations and remainder estimates extract information from limited perturbative data. It surveys a broad spectrum of methods—from Padé approximants and classic acceleration schemes to CNCT and the E-algorithm—and demonstrates their application to both convergence acceleration and resummation of divergent series. The work highlights how Borel summation, often augmented with Padé or conformal-mapping techniques, yields physically meaningful eigenvalues and critical exponents, while also addressing ambiguities arising from Borel-plane singularities and the need for nonperturbative (resurgent) viewpoints. Collectively, the text provides practical algorithms, theoretical foundations, and concrete examples (e.g., Lerch transcendent, incomplete gamma, hydrogen Green function, anharmonic oscillators, and RG analyses) that bridge computational methods with physical predictions, with implications for precision calculations in quantum mechanics and field theory.

Abstract

This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has hitherto not received the attention it actually deserves. The unifying concept for convergence acceleration methods is that in many cases, one can reach much faster convergence than by adding a particular series term by term. In some cases, it is even possible to use a divergent input series, together with a suitable sequence transformation, for the construction of numerical methods that can be applied to the calculation of special functions. This review both aims to provide some practical guidance as well as a groundwork for the study of specialized literature. As a second topic, we review some recent developments in the field of Borel resummation, which is generally recognized as one of the most versatile methods for the summation of factorially divergent (perturbation) series. Here, the focus is on algorithms which make optimal use of all information contained in a finite set of perturbative coefficients. The unifying concept for the various aspects of the Borel method investigated here is given by the singularities of the Borel transform, which introduce ambiguities from a mathematical point of view and lead to different possible physical interpretations. The two most important cases are: (i) the residues at the singularities correspond to the decay width of a resonance, and (ii) the presence of the singularities indicates the existence of nonperturbative contributions which cannot be accounted for on the basis of a Borel resummation and require generalizations toward resurgent expansions. Both of these cases are illustrated by examples.

Figures (5)

• Figure 1: Numerical algorithms used for the Lerch transcendent.
• Figure 2: Numerical evaluation of the $W$ function.
• Figure 3: Numerical evaluation of the $M$ function.
• Figure 4: Exact values (filled circles) for the ground state energy of the quartic anharmonic oscillator as a function of $g$, together with the partial sum of the strong--coupling expansion (solid line) as defined by the first three nonvanishing terms in powers of $g^{-2k/3}$ [see Eq. (\ref{['strong_coupling_expansion']})].
• Figure 5: Parameterization of a four-loop diagram in terms of "triangles" [diagrams (a), (b) and (c)], and various steps in the parameterization of the diagram in Fig. \ref{['haus_diag']}(a) in terms of effective triangle vertices [diagrams (d)--(i)]. Further explanations are given in the text.

Theorems & Definitions (25)

• Definition 1
• Definition 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10