Cauchy's formula on nonempty closed sets and a new notion of Riemann--Liouville fractional integral on time scales

TL;DR

This paper extends Cauchy's formula to arbitrary nonempty closed sets on time scales, leading to new definitions of fractional calculus that unify discrete and continuous cases.

Contribution

It introduces a novel approach to fractional integration and differentiation on time scales based on a generalized Cauchy's formula.

Findings

Cauchy's formula is proven for repeated integration on time scales.

New fractional integral and derivative notions are established for arbitrary closed sets.

The framework unifies continuous and discrete fractional calculus.

Abstract

We prove Cauchy's formula for repeated integration on time scales. The obtained relation gives rise to new notions of fractional integration and differentiation on arbitrary nonempty closed sets.