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arxiv: 2607.01532 · v1 · pith:JW75PHRZnew · submitted 2026-07-01 · 🧮 math.CA

Summability of formal solutions of some singular perturbations problems in differential and moment differential equations

Maria Ksi\k{a}\.zkiewicz , S{\l}awomir Michalik This is my paper

Pith reviewed 2026-07-03 00:46 UTC · model grok-4.3

classification 🧮 math.CA
keywords summabilitysingular perturbationsmoment differential equationsBorel transformformal solutionsCauchy integral formulamultisummability
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The pith

Formal solutions to singularly perturbed linear and moment differential equations are summable under stated assumptions.

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

The paper studies summability for formal solutions of singularly perturbed linear ordinary differential equations and moment differential equations. It concludes that these solutions are summable when the coefficients meet certain conditions, and that the precise type of summability is fixed by the equation. It establishes a link between singularly perturbed moment ordinary differential equations and linear moment partial differential equations, then uses the link to characterize summable and multisummable formal solutions. The proofs rely on Borel transforms, known properties of solutions to moment PDEs, and Cauchy integral representations. A reader would care because summability supplies a concrete function that matches the formal series in a sector, which is the standard way to make sense of divergent series that appear in singular perturbation problems.

Core claim

Under some assumptions the formal solutions of the studied singularly perturbed linear differential and moment differential equations are summable, with the summability type depending on the concrete equation. The connection between singularly perturbed moment ordinary differential equations and linear moment partial differential equations is applied to describe the summable and multisummable formal solutions of the former.

What carries the argument

Borel transforms together with Cauchy integral representations of solutions to associated moment partial differential equations.

If this is right

  • The formal solutions of the examined singularly perturbed equations admit a summability property.
  • The type of summability is determined by the form of each individual equation.
  • The link to moment PDEs yields descriptions of both summable and multisummable formal solutions for the moment ODEs.

Where Pith is reading between the lines

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

  • The same Borel-Cauchy technique could be tested on moment equations whose coefficients violate the current assumptions.
  • The PDE-ODE correspondence may extend to moment equations with variable coefficients of higher order.

Load-bearing premise

The coefficients satisfy conditions that let Borel transforms and Cauchy integral representations be applied directly to the moment differential equations.

What would settle it

An explicit singularly perturbed moment differential equation whose coefficients obey the paper's assumptions yet whose formal solution fails to be summable in the stated sense.

read the original abstract

In this paper we study the summability of solutions of some general forms of singularly perturbed linear ordinary differential and moment differential equations. We conclude that under some assumptions solutions of these equations are summable. The type of this summability depends on the specific equation. We also show the connection between some singularly perturbed moment ordinary differential equations and some linear moment partial differential equations. We apply this connection to describe summable and multisummable formal solutions of these singularly perturbed moment ordinary differential equations. Main techniques used to show these conclusions are based on Borel transforms, properties of solutions of moment partial differential equations and on the Cauchy integral formula together with integral representations of solutions of such equations.

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

2 major / 0 minor

Summary. The paper studies summability of formal solutions to singularly perturbed linear ordinary differential equations and moment differential equations. It concludes that, under unspecified assumptions, the solutions are summable (with the summability type depending on the equation), establishes a connection between singularly perturbed moment ODEs and linear moment PDEs, and applies this to describe summable and multisummable formal solutions. The main techniques invoked are Borel transforms, properties of solutions to moment PDEs, and the Cauchy integral formula with integral representations.

Significance. If the assumptions on coefficients and domains can be made precise and the commutation of Borel transforms with moment operators rigorously justified with explicit growth and sector conditions, the connection between moment ODEs and moment PDEs could provide a useful reduction for analyzing multisummability in singular perturbation problems, extending standard Borel-Laplace techniques to the moment setting.

major comments (2)
  1. [Abstract] Abstract and introduction: the central summability claims are stated only 'under some assumptions' whose precise form (e.g., growth bounds on coefficients, sectorial domains, or conditions on the moment sequence) is never listed or referenced to a specific section; without this, the applicability of the Borel transform and Cauchy representations to the moment operators cannot be verified and the conclusions remain uncheckable.
  2. [Introduction / main results] The reduction from singularly perturbed moment ODEs to moment PDEs via the claimed connection relies on unstated conditions ensuring that the Borel transform commutes with the moment differential operator and that the resulting integral representations remain valid; no growth estimates or domain restrictions are supplied to justify this step, which is load-bearing for the multisummability conclusions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which correctly identify the need for explicit statements of assumptions and conditions. We will revise the manuscript accordingly to address these points.

read point-by-point responses

  1. Referee: [Abstract] Abstract and introduction: the central summability claims are stated only 'under some assumptions' whose precise form (e.g., growth bounds on coefficients, sectorial domains, or conditions on the moment sequence) is never listed or referenced to a specific section; without this, the applicability of the Borel transform and Cauchy representations to the moment operators cannot be verified and the conclusions remain uncheckable.

    Authors: We agree that the assumptions must be stated with precision rather than left implicit. In the revised manuscript we will add an explicit list of all standing assumptions (growth bounds on coefficients, sectorial domains for the Borel transforms, and conditions on the moment sequence) at the end of the introduction, with cross-references to the sections where each assumption is invoked. The abstract will be updated to point to this list. revision: yes

  2. Referee: [Introduction / main results] The reduction from singularly perturbed moment ODEs to moment PDEs via the claimed connection relies on unstated conditions ensuring that the Borel transform commutes with the moment differential operator and that the resulting integral representations remain valid; no growth estimates or domain restrictions are supplied to justify this step, which is load-bearing for the multisummability conclusions.

    Authors: We accept that the commutation property and the validity of the integral representations require explicit justification. The revision will contain a new proposition (placed immediately before the reduction to moment PDEs) that states the precise growth estimates on the coefficients and the sectorial restrictions under which the Borel transform commutes with the moment operator; the same proposition will verify the integral representations via the Cauchy formula under those conditions. revision: yes

Circularity Check

0 steps flagged

No circularity; applies established Borel/moment PDE techniques to singularly perturbed equations without self-referential reduction.

full rationale

The derivation applies Borel transforms, Cauchy integral representations, and known properties of moment PDE solutions to conclude summability of formal solutions for singularly perturbed ODEs and moment differential equations. These steps are presented as direct applications of standard tools under stated assumptions, with the connection between moment ODEs and PDEs serving as an organizational device rather than a definitional loop. No equations reduce by construction to fitted inputs, no uniqueness theorems are imported solely via self-citation, and no ansatz is smuggled through prior author work. The argument remains self-contained against external benchmarks in summability theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, invented entities, or non-standard axioms are mentioned in the abstract; the work rests on standard properties of Borel transforms and Cauchy integrals.

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Reference graph

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