Symmetric duality for left and right Riemann-Liouville and Caputo fractional differences

TL;DR

This paper explores a symmetric duality in discrete fractional calculus, linking left and right Riemann-Liouville and Caputo differences, and derives related summation by parts and Euler-Lagrange equations.

Contribution

It introduces a discrete symmetric duality framework for fractional differences, connecting left and right operators and extending fractional variational calculus.

Findings

Established a symmetric duality relation for fractional differences.

Derived right fractional summation by parts formulas.

Formulated Euler-Lagrange equations for discrete fractional variational problems.

Abstract

A discrete version of the symmetric duality of Caputo-Torres, to relate left and right Riemann-Liouville and Caputo fractional differences, is considered. As a corollary, we provide an evidence to the fact that in case of right fractional differences, one has to mix between nabla and delta operators. As an application, we derive right fractional summation by parts formulas and left fractional difference Euler-Lagrange equations for discrete fractional variational problems whose Lagrangians depend on right fractional differences.