Hyperseries in the non-Archimedean ring of Colombeau generalized numbers.

Abstract

This article is the natural continuation of the paper: Mukhammadiev et al. Supremum, infimum and hyperlimits of Colombeau generalized numbers in this journal. Since the ring of Robinson-Colombeau is non-Archimedean and Cauchy complete, a classical series ${\sum }_{n=0}^{+\infty }{a}_n$ of generalized numbers is convergent if and only if $a_n\to 0$ 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.