Systèmes aux $$q$$-différences singuliers réguliers : classification, matrice de connexion et monodromie. (Regular singular $$q$$-difference systems: Classification, connection matrix and monodromy).(French)Zbl 0957.05012

This paper is connected with the linear $$q$$-difference system $$\sigma_{q}X=AX$$, $$A \in \text{GL}_{n}(C(z))$$, with rational coefficients. Local solutions are constructed in the case of some unresounding system with $$A(0)\in\text{GL}_{n}(C)$$ and in the case of a system with constant coefficients. A global characterization of regular singular systems is given. Next the author studies the behavior for $$q \rightarrow 0$$ of the canonical solutions and for the solutions with a fixed initial value. The behavior of the connection matrix for $$q \rightarrow 0$$ is also investigated. Finally, three examples are given.

MSC:

 05A30 $$q$$-calculus and related topics 39A13 Difference equations, scaling ($$q$$-differences) 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$
Full Text:

References:

 [1] C.R. ADAMS, On the linear ordinary q-difference equations, Ann. Math., série 2, 30, n° 2 (1929), 195-205. · JFM 55.0263.01 [2] C.R. ADAMS, Linear q-difference equations, Bull. Amer. Math. Soc., (1931), 361-399. · JFM 57.0534.05 [3] V.I. ARNOLD, Ordinary differential equations, in Dynamical Systems, Encyclopaedia of Mathematical Sciences, vol. 1, Springer-Verlag, 1980. [4] D. BERTRAND, Groupes algébriques linéaires et théorie de Galois différentielle, Cours 3e cycle, Université Paris VI, 1986. [5] J.-P. BÉZIVIN, Sur LES équations fonctionnelles aux q-différences, Aequationes Math., 43 (1992), 159-176. · Zbl 0757.39002 [6] 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), 521-568. · JFM 44.0391.03 [7] G.D. BIRKHOFF, Formal theory of irregular linear difference equations, Acta Math., 54 (1930), 205-246. · JFM 56.0402.01 [8] J. CANO, J.-P. RAMIS, Théorie de Galois différentielle, livre en préparation, 1999. [9] R.D. CARMICHAEL, The general theory of linear q-difference equations, Amer. J. Math., 34 (1912), 147-168. · JFM 43.0411.02 [10] P. DELIGNE, Équations différentielles à points singuliers réguliers, Lecture Notes in Math., Springer-Verlag, 163 (1970). · Zbl 0244.14004 [11] P.I. ETINGOF, Galois groups and connection matrices of q-difference equations, Electronic Research Announcements of the A.M.S., vol. 1, issue 1 (1995). · Zbl 0844.12004 [12] G. GASPER, M. RAHMAN, Basic hypergeometric series, Encyclopedia of Mathematics, vol. 35, Cambridge University Press, 1990. · Zbl 0695.33001 [13] E.L. INCE, Ordinary differential equations, Dover Publications, 1956. [14] K. IWASAKI, H. KIMURA, S. SHIMOMURA, M. YOSHIDA, From Gauss to Painlevé, Braunschweig, Vieweg, 1991. [15] S. LANG, Elliptic functions, Springer-Verlag, 1987. · Zbl 0615.14018 [16] F. MAROTTE, C. ZHANG, Multisommabilité des séries entières solutions formelles d’une équation aux q-différences linéaire analytique, article en préparation, 1999. · Zbl 1063.39001 [17] D. MUMFORD, Tata lectures on theta, vol I, Birkhäuser, 1983. · Zbl 0509.14049 [18] G. POURCIN (éd.), Rapport du jury de l’agrégation de mathématiques, Ministère de l’Éducation Nationale, Centre National de Documentation Pédagogique, 1994. [19] M. VAN DER PUT, M.F. SINGER, Galois theory of difference equations, Lecture Notes in Math., Springer-Verlag, 1666 (1997). · Zbl 0930.12006 [20] S. RAMANUJAN, Collected works, Chelsea, 1927. [21] J.-P. RAMIS, About the growth of entire functions solutions to linear algebraic q-difference equations, Annales Fac. Sciences de Toulouse, 6, vol. I, n° 1 (1992), 53-94. · Zbl 0796.39005 [22] J.-P. RAMIS, Fonctions θ et équations aux q-différences, non publié, Strasbourg, 1990. [23] J. SAULOY, Matrice de connexion d’un système aux q-différences confluant vers un système différentiel et matrices de monodromie, Preprint, Université Paul Sabatier, Toulouse, 1998. · Zbl 0919.39004 [24] J. SAULOY, Théorie de Galois des équations aux q-différences fuchsiennes, thèse, Université Paul Sabatier, Toulouse, 1999. [25] J. SAULOY, Galois theory of Fuchsian q-differences equations, article en préparation, 2000. [26] L. SCHLESINGER, Handbuch der theorie der linearen differentialgleichungen, Teubner, 1895. · JFM 26.0329.01 [27] W.J. TRJITZINSKY, Analytic theory of linear q-difference equations, Acta Math., 61 (1933), 1-38. · JFM 59.0455.02 [28] W. WASOW, Asymptotic expansions for ordinary differential equations, Dover Publications, 1965. · Zbl 0133.35301 [29] E.T. WHITTAKER, G.N. WATSON, A course of modern analysis, Cambridge University Press, 1927. · JFM 45.0433.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.