Continuous and discrete fractional operators and some fractional functions

TL;DR

This paper introduces fractional extensions of classical orthogonal polynomials using Caputo fractional derivatives and fractional differences, providing new functions with hypergeometric representations and exploring their properties.

Contribution

It presents novel fractional versions of classical orthogonal polynomials via Caputo derivatives and fractional differences, expanding their definitions and properties.

Findings

Defined fractional Hermite, Laguerre, Jacobi, Charlier, Meixner, Krawtchouk, Hahn functions

Provided hypergeometric representations of these fractional functions

Explored properties of the new fractional functions

Abstract

The classical orthogonal polynomials are usually defined by the Rodrigues' formula. This paper refers to a fractional extension of the classical Hermite, Laguerre, Jacobi, Charlier, Meixner, Krawtchouk and Hahn polynomials. By means of the Caputo operator of fractional calculus, C-Hermite, C-Laguerre, C-Legndre and the C-Jacobi functions are defined and their representation in terms of the hypergeometric functions are provided. Also, by means of the Gray and Zhang fractional difference oparator, fractional Charlier, Meixner, Krawtchouk and Hahn functions are defined and their representation in terms of the hypergeometric functions are provided. Some other properties of the new defined functions are given.