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

## TL;DR

This paper investigates the properties of a $q$-analog of Borel-Laplace summation, demonstrating that the product of $q$-summable solutions of certain $q$-difference equations remains $q$-summable and that the summation process acts as a ring morphism.

## Contribution

It proves that the $q$-summation of the product of two $q$-summable series equals the product of their $q$-sums, supporting the conjecture that $q$-summation is a ring morphism.

## Key findings

- The product of two $q$-summable series is $q$-summable.
- The $q$-sum of the product equals the product of the $q$-sums.
- The $q$-summation induces a field morphism under certain conditions.

## 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.