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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
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