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 "Equation missing" of Robinson-Colombeau is non-Archimedean and Cauchy complete, a classical series $$\sum _{n=0}^{+\infty }a_{n}$$ ∑ n = 0 + ∞ a n of generalized numbers "Equation missing" is convergent if and only if $$a_{n}\rightarrow 0$$ a n → 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.

Hyper-power series and generalized real analytic functions

“…iii In [25], we proved that if x is finite, then x n n! s ∈ ρ σ R x is a formal HPS for all gauges ρ, σ .…”

“…In the entire paper, ρ and σ are two arbitrary gauges; only when it will be needed, we will assume a relation between them, such as σ ≤ ρ * or σ ≥ ρ * (see [25]).…”

“…Actually, this (informal) definition does not state explicitly whether a non-convergent series is included or not. However, unlike the real case where finite sums can always be considered, this does not hold for hyperfinite sums in the ring ρ R, see [25]. For this reason, we consider the ρ R-module ρ σ R s of sequences for hyperseries exactly as the space where we can consider hyperfinite sums regardless of convergence.…”

“…For the notation [−] s , see [25,Defi 1]. Elements of ρ σ R x − c are called formal HPS because here we are not considering their convergence.…”

“…In this article, the study of hyperseries in the non-Archimedean ring of Colombeau generalized numbers (CGN), as carried out in [25], is applied to the corresponding notion of hyper-power series. As we will see, this yields results which are more closely related to classical ones, such as, e.g.…”