q-Gevrey asymptotic expansions and Gq-summable series

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. Bezivin, 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.