# Infinitesimal translations and a multivariate Grunwald-Letnikov calclulus

## TL;DR

This paper develops a multivariate generalization of the Grunwald-Letnikov fractional derivative using infinitesimal translations, extending the classical operator and analyzing its properties for multivariate functions.

## Contribution

It introduces a formalism based on infinitesimal translations to justify and extend the Grunwald-Letnikov derivative to multiple variables.

## Key findings

- Constructed a multivariate fractional derivative using a multi-binomial theorem.
- Derived the principal value of the derivative for multivariate power functions.
- Obtained a characteristic equation consistent with recent research.

## Abstract

The goal of this paper is to construct a multivariate generalisation of the Grunwald-Letnikov derivative, a classical fractional derivative operator. To do so, we first produce a formalism of fractional derivatives in terms of infinitesimal translations that justifies the "binomial theorem" argument for the Grunwald-Letnikov derivative, allowing us to then extend the argument to construct the multivariate derivative via the more general multi-binomial theorem. We conclude by studying the principal value of the fractional derivative of a multivariate power function, obtaining a characteristic equation in agreement with recent research in the area.