Local analytic classification of q-difference equations
Astérisque, no. 355 (2013) , 157 p.
@book{AST_2013__355__R1_0,
     author = {Ramis, Jean-Pierre and Sauloy, Jacques and Zhang, Changgui},
     title = {Local analytic classification of $q$-difference equations},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {355},
     year = {2013},
     mrnumber = {3185985},
     zbl = {1320.39008},
     language = {en},
     url = {http://www.numdam.org/item/AST_2013__355__R1_0/}
}
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%A Sauloy, Jacques
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Ramis, Jean-Pierre; Sauloy, Jacques; Zhang, Changgui. Local analytic classification of $q$-difference equations. Astérisque, no. 355 (2013), 157 p. http://numdam.org/item/AST_2013__355__R1_0/

[1] C. Adams - "Linear q-difference equations", Bull. Amer. Math. Soc. 37 (1931), p. 361-400. | MR | Zbl | DOI

[2] Y. André - "Différentielles non commutatives et théorie de Galois différentielles ou aux différences", Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 5, p. 685-739. | MR | Zbl | EuDML | Numdam | DOI

[3] D. Babbitt & V. Varadarajan - Local moduli for meromorphic differential equations, Astérisque, vol. 169-170, Soc. Math. France, 1989. | MR | Zbl | Numdam

[4] B. Berndt & R. Rankin (eds.) - Ramanujan: essays and surveys, Hist. Math., vol. 22, Amer. Math. Soc., 2001. | MR | Zbl | DOI

[5] D. Bertrand - "Extensions panachées et 2-extensions", manuscrit non publié, 1999.

[6] D. Bertrand, "Extensions panachées et dualité", prépublications de l'Institut Mathématique de Jussieu, UMR 287, 2001.

[7] D. Bertrand, "Extensions panachées autoduales", J. K-Theory 11 (2013), p. 393-411. | MR | Zbl

[8] J.-P. Bézivin - "Sur les équations fonctionnelles aux q-différences", Aequationes Math. 43 (1992), no. 2-3, p. 159-176. | MR | Zbl | EuDML | DOI

[9] G. D. Birkhoff - "Singular points of ordinary linear differential equations", Trans. Amer. Math. Soc. 10 (1909), p. 436-470. | MR | JFM | DOI

[10] G. D. Birkhoff, "The generalized Riemann problem for linear differential equations and the allied problems for linear difference and q-difference equations", Proc. Amer. Acad. 49 (1913), p. 521-568. | JFM | DOI

[11] G. D. Birkhoff & P. E. Guenther - "Note on a canonical form for the linear q-difference system", Proc. Nat. Acad. Sci. U.S.A. 27 (1941), p. 218-222. | MR | JFM | DOI

[12] N. Bourbaki - Éléments de mathématique. Algèbre. Chapitre 2: Algèbre linéaire, Hermann, 1947. | MR | Zbl

[13] N. Bourbaki, Éléments de mathématique. Algèbre. Chapitre 10: Algèbre homologique, Masson, 1980. | MR | Zbl

[14] B. Braaksma, B. Faber & G. Immink - "Summation of formal solutions of a class of linear difference equations", Pacific J. Math. 195 (2000), no. 1, p. 35-65. | MR | Zbl | DOI

[15] V. Bugeaud - "Classification analytique et théorie de Galois locales des modules aux q-différences à pentes non entières", C. R. Math. Acad. Sci. Paris 349 (2011), p. 1037-1039. | MR | Zbl | DOI

[16] H. Cartan & S. Eilenberg - Homological algebra, Princeton Math. Ser., vol. 19, Princeton Univ. Press, 1956. | Zbl | MR

[17] R. M. Cohn - Difference algebra, Interscience Publishers John Wiley & Sons, 1965. | MR | Zbl

[18] L. Di Vizio - "Arithmetic theory of q-difference equations: the q-analogue of Grothendieck-Katz's conjecture on p-curvatures", Invent. Math. 150 (2002), no. 3, p. 517-578. | MR | Zbl | DOI

[19] L. Di Vizio, "Local analytic classification of q-difference equations with q=1", J. Noncommut. Geom. 3 (2009), no. 1, p. 125-149. | MR | Zbl | DOI

[20] L. Di Vizio, J.-P. Ramis, J. Sauloy & C. Zhang - "Équations aux q-différences", Gaz. Math. (2003), no. 96, p. 20-49. | MR | Zbl

[21] L. Di Vizio & J. Sauloy - "Outils pour la classification locale des équations aux q-différences linéaires complexes", to appear in L. Di Vizio and T. Rivoal (eds), (Théories galoisiennes et arithmétiques des équations différentielles proceedings of the school held in Luminy, September 2009), Sémin. Congr., 2012. | MR | Zbl

[22] L. Di Vizio & C. Zhang - "On q-summation and confluence", Ann. Inst. Fourier (Grenoble) 59 (2009), no. 1, p. 347-392. | MR | Zbl | EuDML | Numdam | DOI

[23] C. H. Franke - "Picard-Vessiot theory of linear homogeneous difference equations", Trans. Amer. Math. Soc. 108 (1963), p. 491-515. | MR | Zbl | DOI

[24] J. Frenkel - "Cohomologie non abélienne et espaces fibrés", Bull. Soc. Math. France 85 (1957), p. 135-220. | MR | Zbl | EuDML | Numdam | DOI

[25] S. Gelfand & Y. Manin - Methods of homological algebra, 2nd ed., Springer Monogr. Math., Springer, 2003 (transl. from the Russian). | MR | Zbl

[26] A. Grothendieck - "Modèles de Neron et monodromie (avec un appendice par M. Raynaud)", 1972, p. 313-523. | Zbl

[27] P.-F. Hsieh & Y. Sibuya - Basic theory of ordinary differential equations, Universitext, Springer-Verlag, 1999. | MR | Zbl

[28] G. Immink - "On meromorphic equivalence of linear difference operators", Ann. Inst. Fourier (Grenoble) 40 (1990), no. 3, p. 683-699. | MR | Zbl | EuDML | Numdam | DOI

[29] B. Malgrange - "Le théorème de préparation en géométrie différentiable", 1964, exp. 11-13, exp. 22. | Zbl | Numdam

[30] B. Malgrange, Ideals of differentiable functions, Tata Inst. Fund. Res. Stud. Math., vol. 3, 1966. | MR | Zbl

[31]B. Malgrange , "Sur les points singuliers des équations différentielles", Enseignement Math. (2) 20 (1974), p. 147-176. | MR | Zbl

[32] F. Marotte & C. Zhang- "Multisommabilité des séries entières solutions formelles d'une équation aux q-différences linéaire analytique", Ann. Inst. Fourier (Grenoble) 50 (2000), no. 6, p. 1859-1890. | MR | Zbl | EuDML | Numdam | DOI

[33] J. Martinet & J.-P. Ramis - "Problèmes de modules pour des équations différentielles non linéaires du premier ordre", Inst. Hautes Études Sci. Publ. Math. 55 (1982), p. 63-164. | MR | Zbl | EuDML | Numdam | DOI

[34] J. Martins - "Lemme de Watson en dimension supérieure et application à la sommation des solutions des 𝒟-modules holonomes", Ph.D. Thesis, Univ. Paul Sabatier, Toulouse, 2002.

[35] L. Mordell - "The definite integral - e ax 2 +bx e cx+d and the analytic theory of numbers", Acta Math. 61 (1933), p. 323-360. | JFM | Zbl | MR | DOI

[36] O. Ore - "Theory of non-commutative polynomials", Ann. of Math. (2) 34 (1933), no. 3, p. 480-508. | MR | JFM | Zbl | DOI

[37] M. Van Der Put & M. Reversat - "Galois theory of q-difference equations", Ann. Fac. Sci. Toulouse Math. (6) 16 (2007), no. 3, p. 665-718. | MR | Zbl | EuDML | Numdam | DOI

[38] M. Van Der Put & M. F. Singer - Galois theory of difference equations, Lecture Notes in Math., vol. 1666, Springer-Verlag, Berlin, 1997. | MR | Zbl

[39] J.-P. Ramis - "About the growth of entire functions solutions of linear algebraic q-difference equations", Ann. Fac. Sci. Toulouse Math. (6) 1 (1992), no. 1, p. 53-94. | MR | Zbl | EuDML | Numdam | DOI

[40] J.-P. Ramis, Séries divergentes et théories asymptotiques, Bull. Soc. Math. France, vol. 121, 1993, Panor. Synthèses suppl. | MR | Zbl

[41] J.-P. Ramis & J. Sauloy - "The q-analogue of the wild fondamental group. III", accepted for publication by Ann. Sci. Ecole Norm. Sup. in november 2013. | MR

[42] J.-P. Ramis J. Sauloy, "The q-analogue of the wild fundamental group. I", in Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlevé hierarchies, RIMS Kôkyûroku Bessatsu, B2, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007, p. 167-193. | MR | Zbl

[43] J.-P. Ramis J. Sauloy, "The q-analogue of the wild fundamental group. II", in Differential Equations and Singularities, 60th years of J.M. Aroca, vol. 323, Soc. Math. France, 2009. | MR | Zbl | Numdam

[44] J.-P. Ramis, J. Sauloy & C. Zhang - "La variété des classes analytiques d'équations aux q-différences dans une classe formelle", C. R. Math. Acad. Sci. Paris 338 (2004), no. 4, p. 277-280. | MR | Zbl | DOI

[45] J.-P. Ramis, J. Sauloy C. Zhang, "Développement asymptotique et sommabilité des solutions des équations linéaires aux q-différences", C. R. Math. Acad. Sci. Paris 342 (2006), no. 7, p. 515-518. | MR | Zbl | DOI

[46] J.-P. Ramis & C. Zhang - "Développement asymptotique q-Gevrey et fonction thêta de Jacobi", C. R. Math. Acad. Sci. Paris 335 (2002), no. 11, p. 899-902. | MR | Zbl | DOI

[47] J. Roques - "Classification rationnelle et confluence des systèmes aux différences singuliers réguliers", Ann. Inst. Fourier (Grenoble) 56 (2006), no. 6, p. 1663-1699. | MR | Zbl | EuDML | Numdam | DOI

[48] J. Roques, "Sur les équations aux (q-) différences", Ph.D. Thesis, Univ. Paul Sabatier, Toulouse, 2007.

[49] C. Sabbah - "Introduction to algebraic theory of linear systems of differential equations", in Éléments de la théorie des systèmes différentiels. D-modules cohérents et holonomes (Nice, 1990) (P. Maisonobe et al, ed.), Travaux en Cours, vol. 45, Hermann, 1993, p. 1-80. | MR | Zbl

[50] J. Sauloy - "The geometry of the space of isoformal analytic classes of q-difference equations" , in preparation.

[51] J. Sauloy, "Systèmes aux q-différences singuliers réguliers: classification, matrice de connexion et monodromie", Ann. Inst. Fourier (Grenoble) 50 (2000), no. 4, p. 1021-1071. | MR | Zbl | EuDML | Numdam | DOI

[52] J. Sauloy , "Galois theory of Fuchsian q-difference equations", Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 6, p. 925-968 (2004). | MR | Zbl | EuDML | Numdam | DOI

[53] J. Sauloy, "Algebraic construction of the Stokes sheaf for irregular linear q-difference equations", Astérisque 296 (2004), p. 227-251. | MR | Zbl | Numdam

[54] J. Sauloy, "La filtration canonique par les pentes d'un module aux q-différences et le gradué associé", Ann. Inst. Fourier (Grenoble 54 (2004), no. 1, p. 181-210. | MR | Zbl | EuDML | Numdam | DOI

[55] J. Sauloy, "Équations aux q-différences et fibrés vectoriels sur une courbe elliptique * /q 𝐙 ", in Differential Equations and Singularities, 60th years of J.M. Aroca (F. Cano et al., ed.), vol. 323, Soc. Math. France, 2009. | MR | Zbl | Numdam

[56] J.-P. Serre - Algèbre locale. Multiplicités, 2nd ed., Lecture Notes in Math., vol. 11, Springer-Verlag, 1965. | MR | Zbl

[57] V. Tarasov & A. Varchenko - Geometry of q-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Astérisque, vol. 246, Soc. Math. France, 1997. | MR | Zbl | Numdam

[58] W. Trjitzinsky - "Analytic theory of linear q-difference equations", Acta Math. 61 (1933), p. 1-38. | MR | Zbl | JFM | DOI

[59] C. Zhang - "Développements asymptotiques q-Gevrey et séries Gq-sommables", Ann. Inst. Fourier (Grenoble) 49 (1999), no. 1, p. 227-261. | MR | Zbl | EuDML | Numdam | DOI

[60] C. Zhang, "Une sommation discrète pour des équations aux q-différences linéaires et à coefficients analytiques: théorie générale et exemples", in Differential equations and the Stokes phenomenon (B. Braaksma et al., ed.), World Scientific, 2002, p. 309-329. | MR | Zbl | DOI

[61] C. Zhang, "Remarks on some basic hypergeometric series", in Theory and applications of special functions (M. Ismail & E. Koelink, eds.), Dev. Math., vol. 13, Springer, 2005. | MR | Zbl | DOI

[62] C. Zhang, "Solutions asymptotiques et méromorphes d'équations aux q-différences", in Théories asymptotiques et équations de Painlevé (E. Delabaere et al., ed.), Sémin. Congr., vol. 14, Soc. Math. France, 2006, p. 341-356. | MR | Zbl

[63] C. Zhang, "On Mock Theta functions", in preparation, 2010.

[64] C. Zhang, "On the modular behaviour of the infinite product (1-x)(1-xq)(1-xq 2 )(1-xq 3 )...", C. R. Math. Acad. Sci. Paris 349 (2011), p. 725-730. | Zbl | MR | DOI

[65] S. Zhou, Z. Luo & C. Zhang - "On summability of formal solutions to a Cauchy problem and generalization of Mordell's theorem", C. R. Math. Acad. Sci. Paris 348 (2010), no. 13-14, p. 753-758. | MR | Zbl | DOI