On fractional calculus with analytic kernels with respect to functions

TL;DR

This paper introduces and studies a new unified class of fractional calculus operators with analytic kernels with respect to functions, establishing their fundamental properties and connecting two major existing frameworks.

Contribution

It develops a comprehensive theory for fractional integrals and derivatives with analytic kernels with respect to functions, unifying previous approaches and providing new foundational results.

Findings

Established series formulae and composition relations.

Analyzed function spaces and Laplace transforms for the operators.

Unified framework encompasses previous fractional calculus models.

Abstract

Many different types of fractional calculus have been proposed, which can be organised into some general classes of operators. For a unified mathematical theory, results should be proved in the most general possible setting. Two important classes of fractional-calculus operators are the fractional integrals and derivatives with respect to functions (dating back to the 1970s) and those with general analytic kernels (introduced in 2019). To cover both of these settings in a single study, we can consider fractional integrals and derivatives with analytic kernels with respect to functions, which have never been studied in detail before. Here we establish the basic properties of these general operators, including series formulae, composition relations, function spaces, and Laplace transforms. The tools of convergent series, from fractional calculus with analytic kernels, and of operational calculus, from fractional calculus with respect to functions, are essential ingredients in the analysis of the general class that covers both.