# Infinite series representation of fractional calculus: theory and applications

## TL;DR

This paper develops an infinite series framework for fractional calculus, enabling new theoretical insights and practical numerical approximation methods applicable to various definitions and orders of fractional derivatives.

## Contribution

It introduces a universal fractional Taylor series framework that unifies different definitions and orders of fractional calculus, along with an intuitive numerical scheme.

## Key findings

- Validated the effectiveness of the series representation through examples
- Confirmed properties of fractional calculus within the new framework
- Proposed a practical truncation-based numerical approximation method

## Abstract

This paper focuses on the equivalent expression of fractional integrals/derivatives with an infinite series. A universal framework for fractional Taylor series is developed by expanding an analytic function at the initial instant or the current time. The framework takes into account of the Riemann-Liouville definition, the Caputo definition, the constant order and the variable order. On this basis, some properties of fractional calculus are confirmed conveniently. An intuitive numerical approximation scheme via truncation is proposed subsequently. Finally, several illustrative examples are presented to validate the effectiveness and practicability of the obtained results.