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Sauloy The qanalogue of the wild fundamental group (I
 arXiv:math.QA/0611521 v1 17 Nov 2006
"... We describe an explicit construction of galoisian Stokes operators for irregular linear qdifference equations. These operators are parameterized by the points of an elliptic curve minus a finite set of singularities. Taking residues at these singularities, one gets qanalogues of alien derivations ..."
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Cited by 10 (3 self)
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We describe an explicit construction of galoisian Stokes operators for irregular linear qdifference equations. These operators are parameterized by the points of an elliptic curve minus a finite set of singularities. Taking residues at these singularities, one gets qanalogues of alien derivations which freely generate the Lie algebra of the Stokes subgroup of the Galois group. 1
Transseries for a Class of Nonlinear Difference Equations
"... Given a nonlinear analytic difference equation of level 1 with a formal power series solution ^ y 0 we associate with it a stable manifold of solutions with asymptotic expansion ^ y 0 . This manifold can be represented by means of Borel summable series. All solutions with asymptotic expansion ^ y 0 ..."
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Cited by 7 (1 self)
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Given a nonlinear analytic difference equation of level 1 with a formal power series solution ^ y 0 we associate with it a stable manifold of solutions with asymptotic expansion ^ y 0 . This manifold can be represented by means of Borel summable series. All solutions with asymptotic expansion ^ y 0 in some sector can be written as certain exponential series which are called transseries. Some of their properties are investigated: resurgence properties and Stokes transition. Analogous problems for di erential equations have been studied by Costin in [Cos98].
On the generalized RiemannHilbert problem with irregular singularities
 Expo. Math
"... In this article we study the generalized RiemannHilbert problem, which extends the classical RiemannHilbert problem to the case of irregular singularities. The problem is stated in terms of generalized monodromy data which include the monodromy representation, Stokes matrices and the true Poincaré ..."
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Cited by 5 (1 self)
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In this article we study the generalized RiemannHilbert problem, which extends the classical RiemannHilbert problem to the case of irregular singularities. The problem is stated in terms of generalized monodromy data which include the monodromy representation, Stokes matrices and the true Poincaré rank at each singular point. We give sufficient conditions for the existence of a linear differential system with such data. These conditions are in particular fulfilled when the monodromy representation is irreducible, as in the classical case. We solve the problem almost completely in dimension two and three. Our results have applications in differential Galois theory. We give sufficient conditions for a given linear algebraic group G to be the differential Galois group over C(z) of a linear differential system with a minimum number of singularities, all fuchsian but one, at which the system has a minimal Poincaré rank. There are many approaches to differential equations. One can focus on the existence and behaviour of the solutions, or on algebraic properties of their symmetries. One may also
Gbundles, Isomonodromy and Quantum Weyl Groups
 Int. Math. Res. Not
"... It is now twenty years since Jimbo, Miwa and Ueno [23] generalised Schlesinger’s equations (governing isomonodromic deformations of logarithmic connections on vector bundles over the Riemann sphere) to the case of connections with arbitrary order poles. An interesting feature was that new deformatio ..."
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Cited by 4 (1 self)
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It is now twenty years since Jimbo, Miwa and Ueno [23] generalised Schlesinger’s equations (governing isomonodromic deformations of logarithmic connections on vector bundles over the Riemann sphere) to the case of connections with arbitrary order poles. An interesting feature was that new deformation parameters arose: one may vary the ‘irregular
On the analyticity of laguerre series
 Journal of Physics A: Mathematical and Theoretical
, 2008
"... The transformation of a Laguerre series f(z) = ∑ ∞ n=0 λ(α) n L (α) n (z) to a power series f(z) = ∑ ∞ n=0 γnzn is discussed. Since many nonanalytic functions can be expanded in terms of generalized Laguerre polynomials, success is not guaranteed and such a transformation can easily lead to a ..."
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Cited by 3 (1 self)
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The transformation of a Laguerre series f(z) = ∑ ∞ n=0 λ(α) n L (α) n (z) to a power series f(z) = ∑ ∞ n=0 γnzn is discussed. Since many nonanalytic functions can be expanded in terms of generalized Laguerre polynomials, success is not guaranteed and such a transformation can easily lead to a mathematically meaningless expansion containing power series coefficients that are infinite in magnitude. Simple sufficient conditions based on the decay rates and sign patters of the Laguerre series coefficients λ (α) n as n → ∞ can be formulated which guarantee that the resulting power series represents an analytic function. The transformation produces a mathematically meaningful result if the coefficients λ (α) n either decay exponentially or factorially as n → ∞. The situation is much more complicated – but also much more interesting – if the λ (α) n decay only algebraically as n → ∞. If the
Gevrey solutions of threefold singular nonlinear partial differential equations,to appear in
 J. Differential Equations
, 2013
"... We study Gevrey asymptotics of the solutions to a family of threefold singular nonlinear partial differential equations in the complex domain. We deal with both Fuchsian and irregular singularities, and allow the presence of a singular perturbation parameter. By means of the BorelLaplace summation ..."
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We study Gevrey asymptotics of the solutions to a family of threefold singular nonlinear partial differential equations in the complex domain. We deal with both Fuchsian and irregular singularities, and allow the presence of a singular perturbation parameter. By means of the BorelLaplace summation method, we construct sectorial actual holomorphic solutions which turn out to share a same formal power series as their Gevrey asymptotic expansion in the perturbation parameter. This result rests on the MalgrangeSibuya theorem, and it requires to prove that the difference between two neighboring solutions is exponentially small, what in this case involves an asymptotic estimate for a particular Dirichletlike series. Key words: Nonlinear partial differential equations, singular perturbations, formal power series,
Singularly perturbed small step size differencedifferential nonlinear PDEs
 the Journal of Difference Equations
, 2013
"... We study a family of singularly perturbed small step size differencedifferential nonlinear equations in the complex domain. We construct formal solutions to these equations with respect to the perturbation parameter ɛ which are asymptotic expansions with 1−Gevrey order of actual holomorphic solutio ..."
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We study a family of singularly perturbed small step size differencedifferential nonlinear equations in the complex domain. We construct formal solutions to these equations with respect to the perturbation parameter ɛ which are asymptotic expansions with 1−Gevrey order of actual holomorphic solutions on some sectors in ɛ near the origin in C. However, these formal solutions can be written as sums of formal series with a corresponding decomposition for the actual solutions which may possess a different Gevrey order called 1 + −Gevrey in the literature. This phenomenon of two levels asymptotics has been already observed in the framework of difference equations, see [6]. The proof rests on a new version of the socalled RamisSibuya theorem which involves both 1−Gevrey and 1 + −Gevrey orders. Namely, using classical and truncated BorelLaplace transforms (introduced by G. Immink in [18]), we construct a set of neighboring sectorial holomorphic solutions and functions whose difference have exponentially and superexponentially small bounds in the perturbation parameter.
On the parametric Stokes phenomenon for solutions of singularly perturbed linear partial differential equations
 Article ID 930385
"... We study a family of singularly perturbed linear partial differential equations with irregular type (1) in the complex domain. In a previous work [31], we have given sufficient conditions under which the Borel transform of a formal solution to (1) with respect to the perturbation parameter ɛ converg ..."
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We study a family of singularly perturbed linear partial differential equations with irregular type (1) in the complex domain. In a previous work [31], we have given sufficient conditions under which the Borel transform of a formal solution to (1) with respect to the perturbation parameter ɛ converges near the origin in C and can be extended on a finite number of unbounded sectors with small opening and bisecting directions, say κi ∈ [0, 2π), 0 ≤ i ≤ ν − 1 for some integer ν ≥ 2. The proof rests on the construction of neighboring sectorial holomorphic solutions to (1) whose difference have exponentially small bounds in the perturbation parameter (Stokes phenomenon) for which the classical RamisSibuya theorem can be applied. In this paper, we introduce new conditions for the Borel transform to be analytically continued in the larger sectors {ɛ ∈ C ∗ /arg(ɛ) ∈ (κi, κi+1)} where it develops isolated singularities of logarithmic type lying on some half lattice. In the proof, we use a criterion of analytic continuation of the Borel transform described by A. Fruchard and R. Schäfke in [19] and is based on a more accurate description of the Stokes phenomenon for the sectorial solutions mentioned above. Key words: asymptotic expansion, BorelLaplace transform, Cauchy problem, formal power series, integrodifferential equation, linear partial differential equation, singular perturbation, analytic continuation.
Continuous right inverses for the asymptotic Borel map in ultraholomorphic classes via a Laplacetype transform
 Journal of Mathematical Analysis and Applications
, 2012
"... A new construction of linear continuous right inverses for the asymptotic Borel map is provided in the framework of general Carleman ultraholomorphic classes in narrow sectors. Such operators were already obtained by V. Thilliez by means of Whitney extension results for non quasianalytic ultradiffer ..."
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A new construction of linear continuous right inverses for the asymptotic Borel map is provided in the framework of general Carleman ultraholomorphic classes in narrow sectors. Such operators were already obtained by V. Thilliez by means of Whitney extension results for non quasianalytic ultradifferentiable classes, due to J. Chaumat and A. M. Chollet, but our approach is completely different, resting on the introduction of a suitable truncated Laplacetype transform. This technique is better suited for a generalization of these results to the several variables setting. Moreover, it closely resembles the classical procedure in the case of Gevrey classes, so indicating the way for the introduction of a concept of summability which generalizes k−summability theory as developed by J. P. Ramis.
Algorithms for regular solutions of higherorder linear differential systems
 Proceedings of the International Symposium on Symbolic and Algebraic Computations (ISSAC), july 2831, Seoul, Republic of Korea
, 2009
"... We study systems of higherorder linear differential equations having a regular singularity at the origin. Using the properties of matrix polynomials, we develop efficient methods for computing their regular formal solutions. Our algorithm handles the system directly without transforming it into a s ..."
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We study systems of higherorder linear differential equations having a regular singularity at the origin. Using the properties of matrix polynomials, we develop efficient methods for computing their regular formal solutions. Our algorithm handles the system directly without transforming it into a system of firstorder but higher size. We give its arithmetic complexity and discuss the results of an implementation in Maple.