Weyl and Marchaud derivatives: a forgotten history

TL;DR

This paper revisits the historical development of Weyl and Marchaud fractional derivatives, explores their connections with the fractional Laplace operator, and discusses potential generalizations across mathematical fields.

Contribution

It highlights the overlooked contributions of Weyl and Marchaud to fractional calculus and investigates their relationships with the fractional Laplace operator for broader applications.

Findings

Weyl and Marchaud derivatives have a historical significance in fractional calculus.

Connections between fractional Laplace operator and Marchaud derivative are established.

Potential for generalizing these operators to various mathematical fields is discussed.

Abstract

In this paper we recall the contribution given by Hermann Weyl and Andr\'e Marchaud to the notion of fractional derivative. In addition we discuss some relationships between the fractional Laplace operator and Marchaud derivative in the perspective to generalize these objects to different fields of the mathematics.