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43
The qanalogue of the wild fundamental group (I)
, 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 11 (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.
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].
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 6 (2 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 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 6 (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
Monomial summability and doubly singular differential equations
, 2007
"... In this work, we consider systems of differential equations that are doubly singular, i.e. that are both singularly perturbed and exhibit an irregular singular point. If the irregular singular point is at the origin, they have the form εσxr+1 dy dx = f(x, ε,y), f(0, 0,0) = 0 with f analytic in som ..."
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Cited by 6 (0 self)
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In this work, we consider systems of differential equations that are doubly singular, i.e. that are both singularly perturbed and exhibit an irregular singular point. If the irregular singular point is at the origin, they have the form εσxr+1 dy dx = f(x, ε,y), f(0, 0,0) = 0 with f analytic in some neighborhood of (0, 0,0). If the Jacobian dfdy (0, 0,0) is invertible, we show that the unique bivariate formal solution is monomially summable, i.e. summable with respect to the monomial t = εσxr in a (new) sense that will be defined. As a preparation, Poincare ́ asymptotics and Gevrey asymptotics in a monomial are studied.
On q−Gevrey asymptotics for singularly perturbed q−differencedifferential problems with an irregular singularity
, 2011
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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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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
Small divisors and large multipliers
, 2008
"... Nous étudions des germes de champs de vecteurs holomorphes singuliers à l’origine de C n dont la partie linéaire est 1résonante et qui admettent une forme normale polynomiale. En général, bien que le difféomorphisme formel normalisant soit divergent à l’origine, il existe néanmoins des difféomorphi ..."
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Cited by 2 (2 self)
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Nous étudions des germes de champs de vecteurs holomorphes singuliers à l’origine de C n dont la partie linéaire est 1résonante et qui admettent une forme normale polynomiale. En général, bien que le difféomorphisme formel normalisant soit divergent à l’origine, il existe néanmoins des difféomorphismes holomorphes dans des ”domaines sectoriels ” qui les transforment en leur forme normale. Dans cet article, nous étudions la relation qui existe entre le phénomène de petits diviseurs et le caractère Gevrey de ces difféomorphismes sectoriels normalisants. Nous montrons que l’ordre Gevrey de ce dernier est relié au type diophantien des petits diviseurs. We study germs of singular holomorphic vector fields at the origin of C n of which the linear part is 1resonant and which have a polynomial normal form. The formal normalizing diffeomorphism is usually divergent at the origin but there exists holomorphic diffeomorphisms in some ”sectorial domains ” which transform these vector fields into their normal form. In this article, we study the interplay between the small divisors phenomenon and the Gevrey character of the sectorial normalizing diffeomorphisms.
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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Cited by 2 (2 self)
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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.