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

Thomas Dreyfus; Changgui Zhang

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arXiv · 2302.11859

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

Thomas Dreyfus·Changgui Zhang

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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.

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Thomas Dreyfus, Changgui Zhang. 2023-02-23. On the product of two $1$-$q$-summable series. https://arxiv.org/abs/2302.11859

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