On the product of two generic $q$-Gevrey series

Abstract

In this paper, we consider a $q$-analog of the Borel-Laplace summation process introduced by Fabienne Marotte and the second author. We specifically examine two power series solutions of linear $q$-difference equations whose Newton polygon admits only positive slopes equal to $1$. These series, known as the generic $q$-Gevrey series, are shown to have the property that the product of two such series is $Gq$-summable at double level $(1,2)$. Furthermore, we prove that the $Gq$-sum of this product equals the product of the $Gq$-sums of the original two series.