# Analytical Development of Incomplete Riemann-Liouville Fractional Calculus

## TL;DR

This paper develops the theory of incomplete Riemann-Liouville fractional calculus, extending definitions, properties, and operators to analyze a broader class of functions with new mathematical tools.

## Contribution

It introduces rigorous definitions and properties of incomplete Riemann-Liouville fractional differintegrals, including Leibniz rule and composition properties, expanding fractional calculus framework.

## Key findings

- Defined incomplete fractional integrals and derivatives rigorously.
- Proved Leibniz rule for incomplete differintegrals of products.
- Analyzed composition properties with classical calculus operations.

## Abstract

The theory of fractional calculus has developed in a number of directions over the years, including: the formulation of multiple different definitions of fractional differintegration; the extension of various properties of standard calculus into the fractional scenario; the application of fractional differintegrals to assorted special functions. Recently, a new variant of fractional calculus has arisen, namely incomplete fractional calculus. In two very recent papers, incomplete versions of the Riemann-Liouville and Caputo fractional differintegrals have been formulated and applied to several important special functions. In the current work, we develop the theory of incomplete fractional calculus in more depth, investigating further properties of the incomplete Riemann-Liouville fractional differintegrals and answering some fundamental questions about these operators. By considering appropriate function spaces, we formulate rigorously the definitions of incomplete Riemann-Liouville fractional integration, and justify how this model may be used to analyse a wider class of functions than classical fractional calculus. By using analytic continuation, we formulate definitions for incomplete Riemann-Liouville fractional differentiation, hence extending the incomplete integrals to a fully-fledged model of fractional calculus. We also investigate and analyse these operators further, in order to prove new properties. These include a Leibniz rule for incomplete differintegrals of products, and composition properties of incomplete differintegrals with classical calculus operations. These are natural and expected issues to investigate in any new model of fractional calculus, and in the incomplete Riemann-Liouville model the results emerge naturally from the definition previously proposed.