On the product of two $1$-$q$-summable series

Abstract

In this paper we consider a $q$-analog of the Borel-Laplace summation process defined by Marotte and the second author, and consider two series solutions of linear $q$-difference equations with slopes $0$ and $1$. The latter are $q$-summable and we prove that the product of the series is $q$-(multi)summable and its $q$-sum is the product of the $q$-sum of the two series. This is a first step in showing the conjecture that the $q$-summation process is a morphism of rings. We prove that the $q$-summation does induce a morphism of fields by showing that if the inverse of the $q$-Euler series is $q$-summable, then its $q$-sum is not the inverse of the $q$-sum of the $q$-Euler series.