# zbMATH — the first resource for mathematics

Développements asymptotiques $$q$$-Gevrey et séries $$Gq$$-sommables. ($$q$$-Gevrey asymptotic expansions and $$Gq$$-summable series.). (French) Zbl 0974.39009
Summary: We give a $$q$$-analogous version of the Gevrey asymptotics and of the Borel summability respectively due to G. Watson and E. Borel and developed during the last fifteen years by J.-P. Ramis, Y. Sibuya,…The goal of these authors was the study of ordinary differential equations in the complex plane. In the same manner, our goal is the study of $$q$$-difference equations in the complex plane along the way indicated by G. D. Birkhoff and W. J. Trjitzinsky. More precisely, we introduce a new notion of asymptoticity which we call $$q$$-Gevrey asymptotic expansions of order 1. This notion is well adapted to the class of $$q$$-Gevrey power series of order 1 studied by J.-P. Bézivin, J.-P. Ramis and others. Next, we define the class of $$Gq$$-summable power series of order 1 and give a characterization in terms of $$q$$-Borel-Laplace transforms. We show that every power series satisfying a linear analytic $$q$$-difference equation is $$Gq$$-summable of order 1 when the associated Newton polygon has a unique slope equal to 1. We shall study a generalization of this work when the Newton polygon is arbitrary in a later paper.

##### MSC:
 39A13 Difference equations, scaling ($$q$$-differences) 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$ 39B22 Functional equations for real functions 40G99 Special methods of summability
Full Text:
##### References:
 [1] C.R. ADAMS, Linear q-difference equations, Bull. A.M.S., (1931), 361-382. · JFM 57.0534.05 [2] Y. ANDRÉ, Séries Gevrey de type arithmétique (I: théorèmes de pureté et de dualité), preprint, 1997. [3] Y. ANDRÉ, Séries Gevrey de type arithmétique (II: transcendance sans transcendance), preprint, 1997. [4] W. BALSER, B.J.L. BRAAKSMA, J.-P. RAMIS et Y. SIBUYA, Multisummability of formal power series solutions of linear ordinary differential equations, Asymptotic Analysis, 5 (1991), 27-45. · Zbl 0754.34057 [5] J.-P. BÉZIVIN, Sur LES équations fonctionnelles aux q-différences, Aequationes Mathematicae, 43 (1993), 159-176. · Zbl 0757.39002 [6] D.G. BIRKHOFF, The generalized Riemann problem for linear differential equations and the allied problems for linear difference and q-difference equations, Proc. Am. Acad., 49 (1913), 521-568. · JFM 44.0391.03 [7] R.D. CARMICHAEL, The general theory of linear q-difference equations, Am. Jour. Math., 34 (1912), 146-168. · JFM 43.0411.02 [8] M. FLEINERT-JENSEN, Calcul d’indices Gevrey pour des équations aux q-différences, Prépublication de l’IRMA de Strasbourg, 1993. [9] A. FAHIM, J.-P. RAMIS et C. ZHANG, Phénomène de Stokes et groupe de Galois aux q-différences local, en préparation. [10] J.E. LITTLEWOOD, On the asymptotic approximation to integral functions of zero order, Proc. London Math. Soc., Serie 2, no 5 (1907), 361-410. · JFM 38.0450.01 [11] B. MALGRANGE, Sommation des séries divergentes, Expositiones Mathematicae, 13, no 2-3 (1995), 163-222. · Zbl 0836.40004 [12] F. MAROTTE et C. ZHANG, Multisommabilité des séries entières solutions formelles d’une équation aux q-différences linéaire analytique, Prépublication, La Rochelle, 1998. · Zbl 1063.39001 [13] J. MARTINET et J.-P. RAMIS, Elementary acceleration and multisummability I, Ann. Inst. Henri Poincaré, Vol. 54, no 4 (1991), 331-401. · Zbl 0748.12005 [14] J.-P. RAMIS, LES séries k-sommables et leurs applications, Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, Lecture Notes in Physics, 126 (1980), 178-199. [15] J.-P. RAMIS, About the growth of entire functions solutions of linear algebraic q-difference equations, Annales de la Fac. de Toulouse, Série 6, Vol. I, no 1 (1992), 53-94. · Zbl 0796.39005 [16] J.-P. RAMIS, Séries divergentes et théories asymptotiques, Panoramas et synthèses 0, Supplément au Bulletin de la S.M.F., 121 (1993). · Zbl 0830.34045 [17] E.C. TITCHMARSH, The theory of functions, Second edition, Oxford Science Publications, 1939. · Zbl 0022.14602 [18] J.-Cl. TOUGERON, An introduction to the theory of Gevrey expansions and to the Borel-Laplace transform with some applications, Preprint University of Toronto, Canada, 1990. [19] W.J. TRJITZINSKY, Analytic theory of linear q-difference equations, Acta Mathematica, 61 (1933), 1-38. · JFM 59.0455.02 [20] E.T. WHITTAKER et G.N. WATSON, A course of modern analysis, Cambridge Univ. Press, 1927. · JFM 45.0433.02 [21] J. ZENG et C. ZHANG, A q-analog of Newton’s series, Stirling functions and eulerian functions, Results in Math., 25 (1994), 370-391. · Zbl 0816.33010
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.