Arithmetic summable sequence space over non-Newtonian field

TL;DR

This paper introduces new sequence spaces based on arithmetic summability within the framework of geometric calculus over non-Newtonian fields, extending classical summability concepts to this novel setting.

Contribution

It constructs and analyzes the sequence spaces AS(G) and AC(G) for arithmetic summability and convergence in geometric calculus, a new approach over non-Newtonian fields.

Findings

Defined the sequence spaces AS(G) and AC(G) in geometric calculus.

Established properties and relationships of these spaces.

Extended classical summability concepts to non-Newtonian fields.

Abstract

Recently Ruckle \cite{RuckleArithmeticalSummability} introduced the theory of arithmetical summability suggested by the sum $ \sum_{k|m}f(k) $ as $ k $ ranges over the divisors of $m$ including $ 1 $ and $ m .$ Following Ruckle \cite{RuckleArithmeticalSummability} we construct the sequence space $ AS(G) $ and $ AC(G) $ of arithmetic summable and arithmetic convergent sequences in the sense of geometric calculus and derive interesting results in the geometric field.