Hyperseries in the non-Archimedean ring of Colombeau generalized numbers

TL;DR

This paper introduces hyperseries in the non-Archimedean Colombeau ring to extend classical convergence concepts, enabling the study of analytic generalized functions and classical results within this framework.

Contribution

It proposes the notion of hyperseries to generalize convergence in the non-Archimedean Colombeau ring, overcoming limitations of classical series convergence.

Findings

Hyperseries recover classical analytic functions in the Colombeau setting

Classical results are extended to the non-Archimedean context

Convergence properties are characterized using hyperseries

Abstract

This article is the natural continuation of the paper: Mukhammadiev A.~et al Supremum, infimum and hyperlimits of Colombeau generalized numbers in this journal. Since the ring $\tilde{R}$ of Robinson-Colombeau is non-Archimedean, a classical series $\sum_{n=0}^{+\infty}a_{n}$ of generalized numbers $a_{n}\in\tilde{R}$ is convergent if and only if $a_{n}\to0$ in the sharp topology. Therefore, this property does not permit us to generalize several classical results, mainly in the study of analytic generalized functions (as well as, e.g., in the study of sigma-additivity in integration of generalized functions). Introducing the notion of hyperseries, we solve this problem recovering classical examples of analytic functions as well as several classical results.