Multisummability of formal power series solutions of linear ordinary differential equations

Balser, Werner; Braaksma, B. L. J.; Ramis, Jean-Pierre; Sibuya, Yasutaka

doi:10.3233/asy-1991-5102

Cited by 77 publications

(43 citation statements)

References 3 publications

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“…A convergent one-parameter representation of tronqueé solutions. We normalize (1) as described in [11]: the following refinement of the Boutroux substitution z = 24 −1 30 4/5 x 4/5 e −πi/5 ; y(z) = i z/6(1 − 4 25 x −2 + h(x)) (5) where the branch of the square root is the usual one, which is positive for z > 0, brings (1) to the Boutroux-like form…”

Section: 2mentioning

confidence: 99%

Tronquée Solutions of the Painlevé Equation PI

Costin

Huang

2015

Constr Approx

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We analyze the one parameter family of tronquée solutions of the Painlevé equation PI in the pole-free sectors together with the region of the first array of poles. We find a convergent expansion for these solutions, containing one free parameter multiplying exponentially small corrections to the Borel summed power series. We link the position of the poles in the first array to the free parameter, and find the asymptotic expansion of the pole positions in this first array (in inverse powers of the independent variable). We show that the tritronquées are given by the condition that the parameter be zero. We show how this analysis in conjunction with the asymptotic study of the pole sector of the tritronquée in [12] leads to a closed form expression for the Stokes multiplier directly from the Painlevé property, not relying on isomonodromic or related type of results.

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Section: 2mentioning

confidence: 99%

Tronquée Solutions of the Painlevé Equation PI

Costin

Huang

2015

Constr Approx

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“…There are various ways to construct Σ i ( F ), although the details will not be needed here. In particular the series F is '(k − 1)-summable' on Sect i , with sum Σ i ( F )-see [10,48,50]. Other approaches appear in [11,43].…”

Section: The Anti-stokes Directionsmentioning

confidence: 99%

Symplectic Manifolds and Isomonodromic Deformations

Boalch¹

2001

Advances in Mathematics

166

271

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Definition 2.5. The moduli space M * (a) is the set of isomorphism classes of pairs (V, ∇) where V is a trivial rank n holomorphic vector bundle over P 1 and ∇ is a meromorphic connection on V which is formally equivalent to d − i A 0 at a i for each i and has no other poles.Following [40], we also define slightly larger moduli spaces:Definition 2.6. The extended moduli space M * (a) is the set of isomorphism classes of triples (V, ∇, g) consisting of a generic connection ∇ (with poles on D) on a trivial holomorphic vector bundle V over P 1 with compatible framings g = ( 1 g 0 , . . . , m g 0 ) such that (V, ∇, g) has irregular type i A 0 at each a i .The term 'extended moduli space' is taken from the paper [37] of L.Jeffrey, since these spaces play a similar role (but are not the same).Remark 2.1. For use in later sections we also define spaces M(a) and M(a) simply by replacing the word 'trivial' by 'degree zero' in Definitions 2.5 and 2.6 respectively.

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“…These series turn out to be Gevrey asymptotic representations of actual solutions defined in suitable domains, and there is the possibility of reconstructing such analytic solutions from the formal ones by a process known as multisummability (in a sense, an iteration of a finite number of elementary k−summability procedures), developed in the 1980's by J.-P. Ramis, J.Écalle, W. Balser et al This technique has been proven to apply successfully to a plethora of situations concerning the study of formal power series solutions at a singular point for linear and nonlinear (systems of) meromorphic ordinary differential equations in the complex domain (see, to cite but a few, the works [1,3,11,38,49]), for partial differential equations (for example, [2,4,16,35,36,44,54]), as well as for singular perturbation problems (see [5,13,26], among others).…”

Section: Introductionmentioning

confidence: 99%

Strongly regular sequences and proximate orders

Jiménez-Garrido

Sanz

2016

Journal of Mathematical Analysis and Applications

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Summability methods for ultraholomorphic classes in sectors, defined in terms of a strongly regular sequence (M p ) p∈N0 , have been put forward by A. Lastra, S. Malek and the second author [27]. We study several open questions related to the existence of kernels of summability constructed by means of analytic proximate orders. In particular, we give a simple condition that allows us to associate a proximate order with a strongly regular sequence. Under this assumption, and through the characterization of strongly regular sequences in terms of so-called regular variation, we show that the growth index γ(M) defined by V.Thilliez [55] and the order of quasianalyticity ω(M) introduced by the second author [50] are the same.

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Multisummability of formal power series solutions of linear ordinary differential equations

Cited by 77 publications

References 3 publications

Tronquée Solutions of the Painlevé Equation PI

Tronquée Solutions of the Painlevé Equation PI

Symplectic Manifolds and Isomonodromic Deformations

Strongly regular sequences and proximate orders

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