On the summability of formal power series solutions of \(q\)-difference equations. I.
(Sur la sommabilité des séries entières solutions formelles d’une équation aux \(q\)-différences. I.)

*(French)*Zbl 0913.39002Summary: We give a \(q\)-analogous version of the Gevrey asymptotic 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\(\dots\) 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 [Acta Math. 61, 1-38 (1933; Zbl 0007.21103)].

More precisely, we introduce a new notion of asymptoticality 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. 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 unic slope equal to 1. We shall study a generalization of this work when the Newton polygon is arbitrary in a later paper.

More precisely, we introduce a new notion of asymptoticality 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. 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 unic slope equal to 1. We shall study a generalization of this work when the Newton polygon is arbitrary in a later paper.

##### MSC:

39A10 | Additive difference equations |

30B10 | Power series (including lacunary series) in one complex variable |

40A30 | Convergence and divergence of series and sequences of functions |