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arxiv: 2605.06687 · v1 · submitted 2026-04-29 · 🧮 math.GM

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Asymptotic Convergence of Weniger's δ-Transformation for a Class of Superfactorially Divergent Stieltjes Series

Riccardo Borghi
Authors on Pith no claims yet

Pith reviewed 2026-05-11 01:18 UTC · model grok-4.3

classification 🧮 math.GM
keywords Weniger δ-transformationStieltjes seriessuperfactorial divergenceasymptotic convergencetruncation errorintegral representationconvergence rateresummation
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The pith

Weniger's δ-transformation converges asymptotically for superfactorially divergent Stieltjes series with (2n)! moment growth.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper addresses the resummation of Stieltjes series whose moment sequence grows as (2n)!, a class that diverges superfactorially yet satisfies Carleman's condition. Padé approximants converge slowly because the associated Carleman series diverges only logarithmically. Weniger's δ-transformation is advanced as a more efficient alternative. By applying established results on converging factors, the author obtains an exact integral representation of the truncation error. From this representation the leading-order asymptotic behavior of the error and the associated convergence rate are derived rigorously for real positive arguments, with numerical experiments confirming the analysis.

Core claim

For the class of Stieltjes series with moment sequence growing as (2n)!, Weniger's δ-transformation admits an exact integral representation of its truncation error. This representation yields the leading-order asymptotic behavior of the transformation error together with an estimate of the convergence rate for real positive arguments.

What carries the argument

Exact integral representation of the truncation error for Weniger's δ-transformation, obtained by direct application of converging-factor results for superfactorially divergent Stieltjes series.

If this is right

  • The δ-transformation error follows a definite leading asymptotic form for real positive arguments.
  • An explicit estimate of the convergence rate becomes available.
  • The method supplies a computationally efficient alternative to Padé approximants for this class of series.
  • Numerical experiments confirm the derived asymptotic predictions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The integral-representation technique may transfer to the error analysis of other nonlinear transformations applied to similar divergent series.
  • The same framework could be tested on series with different moment growth rates to map out the range of applicability.
  • Results for positive real arguments provide a foundation for exploring analytic continuation to complex values.

Load-bearing premise

Established results on converging factors for superfactorially divergent Stieltjes series apply directly to produce an exact integral representation of the truncation error specifically for Weniger's δ-transformation.

What would settle it

Numerical evaluation of the actual truncation error for a concrete series with moments proportional to (2n)! at successively higher transformation orders that deviates from the predicted leading asymptotic term would disprove the derivation.

Figures

Figures reproduced from arXiv: 2605.06687 by Riccardo Borghi.

Figure 1
Figure 1. Figure 1: FIG. 1: Comparison of the observed transformation errors (open circles) vs the transformation order [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The same as in Fig. 1, but for complex [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Behaviours of Pad´e approximant sequences [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Behaviour of the relative error, against the transformation order [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Behaviour of the relative error, against the transformation order [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
read the original abstract

The resummation of superfactorially divergent series represents a significant computational challenge in mathematical physics. In the present paper the resummation of a specific class of Stieltjes series characterized by a moment sequence growing as $(2n)!$ will be addressed. Despite the fact that Carleman's condition is satisfied for these series, the convergence rate of Pad\'e approximants is severely hindered by the logarithmic divergence of the associated Carleman series. Weniger's $\delta$ transformation is proposed as a highly efficient alternative resummation tool. By employing recently established results on the converging factors of superfactorially divergent Stieltjes series, an exact integral representation for the truncation error is obtained. This representation enables the rigorous derivation of the leading-order asymptotic behavior of the transformation error, as well as the estimation of the related convergence rate, for real positive arguments. Numerical experiments strongly support the theoretical findings, suggesting that the $\delta$ transformation offers a robust and computationally efficient framework for decoding this class of wildly divergent expansions

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper addresses the resummation of superfactorially divergent Stieltjes series whose moments grow as (2n)!. It proposes Weniger's δ-transformation as an efficient alternative to Padé approximants, whose convergence is hindered by the logarithmic divergence of the associated Carleman series. By invoking recently established results on converging factors, the authors obtain an exact integral representation of the truncation error; this representation is then used to derive the leading-order asymptotic behavior of the transformation error and the associated convergence rate for positive real arguments. Numerical experiments are presented in support of the theoretical predictions.

Significance. If the derivation holds, the work supplies a rigorous, parameter-free asymptotic analysis for a resummation method applied to a class of series that arise in mathematical physics and for which standard Padé methods converge slowly. The reliance on external converging-factor results keeps the construction free of ad-hoc fitting, while the numerical experiments furnish falsifiable predictions of the convergence rate; both features strengthen the practical utility of the claimed result.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'a specific class of Stieltjes series characterized by a moment sequence growing as (2n)!' would benefit from an explicit one-line definition of the underlying Stieltjes integral or generating function so that readers can immediately identify the precise series under study.
  2. [Numerical experiments] Numerical experiments section: the abstract states that the experiments 'strongly support' the theory, but does not indicate the range of positive real arguments tested or the observed scaling of the error; adding these quantitative details would make the corroboration easier to assess.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation applies external results to derive new asymptotics

full rationale

The paper obtains an exact integral representation for the truncation error of Weniger's δ-transformation by invoking recently established results on converging factors of (2n)!-moment Stieltjes series. From this representation it derives the leading-order asymptotic error behavior and convergence rate for positive real arguments. No load-bearing step reduces the claimed asymptotics to a quantity defined inside the paper by construction, self-fit, or self-citation chain; the construction is parameter-free, the cited results are treated as independent input, and numerical experiments are presented that can falsify the predicted rate. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of Stieltjes series and on previously published results about converging factors; no new free parameters are introduced and no new entities are postulated.

axioms (2)
  • domain assumption The series under consideration is a Stieltjes series whose moment sequence grows as (2n)!
    Explicitly stated as the class addressed in the abstract.
  • standard math Carleman's condition is satisfied while the associated Carleman series diverges only logarithmically
    Invoked to explain why Padé approximants are hindered.

pith-pipeline@v0.9.0 · 5478 in / 1506 out tokens · 53130 ms · 2026-05-11T01:18:42.845630+00:00 · methodology

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Reference graph

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