Axiomatic Local Metric Derivatives for Low-Level Fractionality with Mittag-Leffler Eigenfunctions

TL;DR

This paper introduces an axiomatic local metric derivative with Mittag-Leffler eigenfunctions suitable for low-level fractionality, providing a potentially more rigorous alternative to existing local fractional derivatives.

Contribution

It develops a new axiomatic local metric derivative that is valid near order 1 and offers a solid foundation compared to previous approaches based on Jumarie.

Findings

The derivative has Mittag-Leffler eigenfunctions.

It is valid for low-level fractionality near order 1.

Provides an axiomatic basis for local fractional derivatives.

Abstract

In this contribution, we build up an axiomatic local metric derivative that exhibits the Mittag-Leffler as an eigenfunction and is valid for low-level fractionality, whenever the order parameter is close to $1$. This version of deformed or metric derivative may be a possible alternative to the versions by Jumarie and the inappropriately so-called local fractional derivative also based on the Jumarie's approach. With rules similar to the classical ones, but with a solid axiomatic basis in the limit pointed out here, we present our results and some comments on the limits of validity for the controversial formalism found in the literature of the area.