## 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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### References:

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