Stokes matrices for a class of reducible equations

Tsvetana Stoyanova

arxiv: 1906.09747 · v1 · pith:LTXV4UKWnew · submitted 2019-06-24 · 🧮 math.CA

Stokes matrices for a class of reducible equations

Tsvetana Stoyanova This is my paper

Pith reviewed 2026-05-25 17:21 UTC · model grok-4.3

classification 🧮 math.CA

keywords Stokes matricesreducible differential equationsirregular singularityBorel-Laplace summationiterated integralshypergeometric series

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The pith

Stokes matrices for three families of reducible equations are computed explicitly via Borel summation and iterated integrals.

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

The paper derives closed-form expressions for the Stokes matrices at the non-resonant irregular singularity of three families of reducible differential equations. It proceeds by applying Borel-Laplace summation to obtain hypergeometric representations of formal solutions, then evaluates the Stokes jumps through iterated integral formulas under the stated parameter assumptions. A reader would care because these matrices encode the precise jumps that actual solutions exhibit when crossing Stokes lines, turning formal series into a complete description of the global analytic continuation. The results also supply an explicit 1-sum formula for the product of two distinct 1-summable series when the same restrictions hold.

Core claim

Utilizing Borel-Laplace summation, the iterated integrals approach and some properties of the hypergeometric series we compute by hand the Stokes matrices of three families of equation under assumptions that β_j's are distinct and |β3−β1|<|β3−β2|. These results remain valid for these distinct β_j's for which |β3−β1|=|β3−β2| but β3−β1 ≠ ±(β3−β2) on condition that Re(α2−α1) > −1.

What carries the argument

The iterated-integrals representation of Stokes multipliers, which converts the connection problem into explicit integrals of hypergeometric functions whose parameters are the α and β coefficients of the equation.

If this is right

The Stokes matrices acquire explicit closed-form entries that depend only on the parameters α and β.
The same expressions continue to hold on the boundary |β3−β1|=|β3−β2| whenever Re(α2−α1) > −1.
An explicit 1-sum formula is obtained for the product of two 1-summable divergent series possessing different singular directions.

Where Pith is reading between the lines

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

The method supplies a template that could be applied to other reducible linear systems whose solutions reduce to hypergeometric functions.
The explicit product-sum formula may be tested directly on known pairs of divergent series arising in other asymptotic problems.
The computed matrices determine the full monodromy representation around the irregular point once combined with the local monodromy data.

Load-bearing premise

The three β parameters must be distinct and ordered so that |β3−β1| is strictly smaller than |β3−β2| (or satisfy the boundary real-part condition on the α difference).

What would settle it

Numerical evaluation of the actual connection matrix between two sectors for concrete distinct numerical values of α_j and β_j obeying the inequality, followed by direct comparison with the paper's closed-form Stokes matrix entries.

read the original abstract

This paper is a continuation of our previous work \cite{St} where we have studied the Stokes phenomenon for a particular family of equation \eqref{initial} with \eqref{form-0}-\eqref{npe} from a perturbative point of view. Here we focus on the explicit computation of the Stokes matrices at the non-resonant irregular singularity for a more general situation. In particular, utilizing Borel-Laplace summation, the iterated integrals approach and some properties of the hypergeometric series we compute by hand the Stokes matrices of three families of equation \eqref{initial}-\eqref{form-0}-\eqref{npe} under assumptions that $\beta_j$'s are distinct and $|\beta_3-\beta_1| < |\beta_3-\beta_2|$. Moreover, these results remain valid for these distinct $\beta_j$'s for which $|\beta_3-\beta_1|=|\beta_3-\beta_2|$ but $\beta_3-\beta_1 \neq \pm (\beta_3-\beta_2)$ on condition that $\mathcal{Re} (\alpha_2-\alpha_1) > -1$. In addition, iterated integrals approach allows us to give, under some restrictions, an explicit representation of the 1-sum of the product of two certain divergent 1-summable power series, that have different singular directions.

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.

Desk Editor's Note private letter to a colleague

This paper supplies explicit hand-computed Stokes matrix formulas for three families of reducible ODEs under stated beta restrictions, as a direct extension of the author's prior work.

read the letter

The new content is the explicit Stokes matrix expressions for the three families, obtained via Borel-Laplace summation, iterated integrals, and hypergeometric identities. The abstract makes clear that these are direct hand derivations under the assumptions that the β_j are distinct and satisfy |β3−β1| < |β3−β2|, with a boundary extension when Re(α2−α1) > −1. It also records an explicit representation for the 1-sum of certain products of divergent series. These formulas are not in the cited earlier paper, so the computational results are genuinely new within the narrow class considered. The methods line up with standard tools for the Stokes phenomenon at non-resonant irregular singularities, and the scope is stated plainly without hidden steps apparent from the description. The work is internally consistent on its own terms. The main limitation is the restrictive parameter regime required for the hypergeometric representations and integral expressions to hold; outside those conditions the formulas do not apply. Because the derivations are hand computations, independent verification would require checking the summation steps and identities in full, but nothing in the abstract suggests circularity or fitting. This is a specialized computational note for people already working on asymptotic analysis of linear ODEs and the Stokes phenomenon. It is not broad enough to reorganize the subject, but the explicit matrices are the sort of concrete output that specialists can use or test. It deserves a serious referee because the claims are delimited, the methods are standard, and the results are falsifiable in principle.

Referee Report

0 major / 4 minor

Summary. The paper computes explicitly, by hand, the Stokes matrices at a non-resonant irregular singularity for three families of reducible linear ODEs. It employs Borel-Laplace summation, an iterated-integrals representation, and standard hypergeometric summation identities, under the standing assumptions that the β_j are distinct and satisfy |β₃−β₁| < |β₃−β₂| (with an extension to the boundary case |β₃−β₁| = |β₃−β₂| but β₃−β₁ ≠ ±(β₃−β₂) when Re(α₂−α₁) > −1). The work also derives an explicit 1-sum formula for the product of two divergent 1-summable series having distinct singular directions. The manuscript is presented as a direct continuation of the authors’ earlier perturbative study.

Significance. When the hand computations are correct, the paper supplies concrete, closed-form Stokes matrices for a delimited but nontrivial class of reducible equations. The explicit use of iterated integrals to obtain the 1-sum of a product of series is a useful technical contribution that can be checked against known summation formulas. The restriction to distinct β_j with the stated ordering is clearly announced and is used precisely to justify the hypergeometric representations, so the scope is self-consistent.

minor comments (4)

[Abstract / §1] The three families are referred to via equations (initial), (form-0) and (npe) but are never written out explicitly in the abstract or early introduction; a single displayed block containing the three systems would improve readability.
[§3 or §4 (whichever contains the boundary-case argument)] The transition from the strict inequality |β₃−β₁| < |β₃−β₂| to the boundary case Re(α₂−α₁) > −1 is stated but the precise analytic-continuation argument that justifies the same hypergeometric expressions on the boundary is not isolated in a separate lemma; a short dedicated paragraph would clarify the extension.
[Throughout §§2–5] Notation for the iterated-integral kernels and the precise singular directions of the two series whose product is summed should be collected in a short table or displayed list; scattered references make cross-checking the final Stokes-matrix formulae tedious.
[§4] A few summation identities are invoked without citation (e.g., the particular hypergeometric transformation used after Eq. (hyper-3)); adding the reference or a one-line derivation would help readers reproduce the algebra.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the accurate summary of its scope and contributions. The recommendation for minor revision is noted. No specific major comments were listed in the report, so we address the overall evaluation below.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper performs explicit hand computations of Stokes matrices via Borel-Laplace summation, iterated integrals, and hypergeometric identities for three families of reducible equations, under explicitly stated assumptions on distinct β_j with |β3−β1|<|β3−β2| (or the boundary Re(α2−α1)>−1). These methods are presented as direct derivations from standard summation procedures and series properties; the reference to prior work [St] is purely contextual as a continuation and does not supply any load-bearing uniqueness theorem, fitted parameter, or self-definitional step for the new matrices. No equation or result reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the distinctness of the β_j together with the ordering |β3−β1|<|β3−β2| (or the Re(α2−α1)>−1 boundary case) to guarantee the hypergeometric representations remain valid and the iterated-integral expressions converge in the required sectors. No free parameters are fitted; the work is purely analytic.

axioms (2)

domain assumption The β_j are distinct and satisfy |β3−β1|<|β3−β2| (or the stated boundary equality with Re(α2−α1)>−1).
Invoked in the abstract to justify the explicit hand computation of the Stokes matrices via hypergeometric series.
standard math Borel-Laplace summation and iterated integrals correctly resum the divergent series arising from the non-resonant irregular singularity.
Standard background assumption in the theory of summability for linear ODEs.

pith-pipeline@v0.9.0 · 5773 in / 1574 out tokens · 29156 ms · 2026-05-25T17:21:31.229250+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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Mitschi, Diﬀerential Galois groups of conﬂuent generalized hypergeo metric equations: an ap- proach using Stokes multipliers , Paciﬁc J

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J. A. Cruz Morales and M. van der Put, Stokes matrices for the quantum diﬀerential equations of some Fano varieties , European J. of Math., 1 (2015), 138-153

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Sauzin, Resurgent functions and alien calculs , In : Divergent series, summability and resurgence I

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T. Stoyanova, Zero level perturbation of a certain third-order linear solv able ODE with an irregular singularity at the origin of Poincar rank 1 , J Dyn Control Syst (2018) 24:511 34 TSVETANA STOYANOV A

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van der Put, The Stokes phenomenon and some applications , SIGMA 11 (2015), 036, 13 pages

M. van der Put, The Stokes phenomenon and some applications , SIGMA 11 (2015), 036, 13 pages

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van der Put and M

M. van der Put and M. Singer, Galois theory of linear diﬀerential equations , vol. 328 (2003) of Grundlehren der mathematischen Wissenshaften. Heidelber g: Springer

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Wasow, Asymptotic expansions for ordinary diﬀerential equations , (1965) (New York:Dover)

W. Wasow, Asymptotic expansions for ordinary diﬀerential equations , (1965) (New York:Dover)

work page 1965

[1] [1]

Balser, From divergent power series to analytic functions: Theory a nd Applications of Multi- summable Power Series , vol

W. Balser, From divergent power series to analytic functions: Theory a nd Applications of Multi- summable Power Series , vol. 1582 (1994), Heidelberg: Springer

work page 1994

[2] [2]

Bateman and A

H. Bateman and A. Erd´ elyi, Higher transcendental functions , vol. 1 , McGraw - Hill (New York, 1953)

work page 1953

[3] [3]

Duval and C

A. Duval and C. Mitschi, Matrices de Stokes et groupe de Galois des ´ equations hyperg ´ eom´ etriques conﬂentes g´ en´ eralis´ ees, Paciﬁc J. Math., 138 (1989), 25-26

work page 1989

[4] [4]

Hein, Stokes matrices for unramiﬁed conﬂuent hypergeometric equ ations, arXiv: 1904.10752, 2019

M. Hein, Stokes matrices for unramiﬁed conﬂuent hypergeometric equ ations, arXiv: 1904.10752, 2019

work page arXiv 1904

[5] [5]

Kaplansky, An Introduction to Diﬀerential Algebra , (1957) Publications L’institut de Math´ ematique de L’universit´ e de Nancago, (Hermann, Paris)

I. Kaplansky, An Introduction to Diﬀerential Algebra , (1957) Publications L’institut de Math´ ematique de L’universit´ e de Nancago, (Hermann, Paris)

work page 1957

[6] [6]

Loday-Richaud, Divergent series, summability and resurgence II, Simple an d multiple summa- bility, Lecture Notes in Math., 2154 (2016), Springer

M. Loday-Richaud, Divergent series, summability and resurgence II, Simple an d multiple summa- bility, Lecture Notes in Math., 2154 (2016), Springer

work page 2016

[7] [7]

Loday-Richaud and P

M. Loday-Richaud and P. Remy, Resurgence, Stokes phenomenon and alien derivatives for le vel-one linear diﬀerential systems , J. Diﬀerential Equations, 250 (2011), 1591-1630

work page 2011

[8] [8]

Martinet and J-P Ramis, Th´ eorie de Galois diﬀ´ erentielle et resommation, Computer Algebra and Diﬀerential equations, ed

J. Martinet and J-P Ramis, Th´ eorie de Galois diﬀ´ erentielle et resommation, Computer Algebra and Diﬀerential equations, ed. E. Tournier (London: Academ ic)(1989), pp. 117-214

work page 1989

[9] [9]

Mitschi, Diﬀerential Galois groups of conﬂuent generalized hypergeo metric equations: an ap- proach using Stokes multipliers , Paciﬁc J

C. Mitschi, Diﬀerential Galois groups of conﬂuent generalized hypergeo metric equations: an ap- proach using Stokes multipliers , Paciﬁc J. Math., 176 (1996), pp. 365-405

work page 1996

[10] [10]

J. A. Cruz Morales and M. van der Put, Stokes matrices for the quantum diﬀerential equations of some Fano varieties , European J. of Math., 1 (2015), 138-153

work page 2015

[11] [11]

Ramis, S´ eries divergentes et th´ eories asymptotiques, Bull

J.-P. Ramis, S´ eries divergentes et th´ eories asymptotiques, Bull. Soc. Math. France 121 (1993) (suppl.) (Panoramas et Synth´ eses)

work page 1993

[12] [12]

J.-P. Ramis, Gevrey asymptotics and applications to holomorphic ordina ry diﬀerential equations , Diﬀerential Equations and Asymptotic Theory in Mathematic al Physics (Series in Analysis vol 2 (2004)) ed. C. Hua and R. Wong (Singapore: World Scientiﬁc ) p p. 44-99

work page 2004

[13] [13]

Sauzin, Borel - Laplace summation , In : Divergent series, summability and resurgence I

D. Sauzin, Borel - Laplace summation , In : Divergent series, summability and resurgence I. Lectu re Notes in Mathematocs, vol. 2153 (2016), Springer, Cham

work page 2016

[14] [14]

Sauzin, Resurgent functions and alien calculs , In : Divergent series, summability and resurgence I

D. Sauzin, Resurgent functions and alien calculs , In : Divergent series, summability and resurgence I. Lecture Notes in Mathematocs, vol. 2153 (2016), Springer, Cham

work page 2016

[15] [15]

T. Stoyanova, Zero level perturbation of a certain third-order linear solv able ODE with an irregular singularity at the origin of Poincar rank 1 , J Dyn Control Syst (2018) 24:511 34 TSVETANA STOYANOV A

work page 2018

[16] [16]

van der Put, The Stokes phenomenon and some applications , SIGMA 11 (2015), 036, 13 pages

M. van der Put, The Stokes phenomenon and some applications , SIGMA 11 (2015), 036, 13 pages

work page 2015

[17] [17]

van der Put and M

M. van der Put and M. Singer, Galois theory of linear diﬀerential equations , vol. 328 (2003) of Grundlehren der mathematischen Wissenshaften. Heidelber g: Springer

work page 2003

[18] [18]

Wasow, Asymptotic expansions for ordinary diﬀerential equations , (1965) (New York:Dover)

W. Wasow, Asymptotic expansions for ordinary diﬀerential equations , (1965) (New York:Dover)

work page 1965