On delta and nabla Caputo fractional differences and dual identities

TL;DR

This paper explores dual identities for Caputo fractional differences, introduces new types of these differences, and investigates their relations, including applications to fractional difference equations and integration by parts formulas.

Contribution

It introduces a dual type of Caputo fractional differences that satisfy specific dual identities and explores their relations with Riemann differences and Mittag-Leffler functions.

Findings

Established dual identities for Caputo fractional differences.

Connected Caputo and Riemann fractional differences.

Solved fractional difference equations using Mittag-Leffler functions.

Abstract

We Investigate two types of dual identities for Caputo fractional differences. The first type relates nabla and delta type fractional sums and differences. The second type represented by the Q-operator relates left and right fractional sums and differences. Two types of Caputo fractional differences are introduced, one of them (dual one) is defined so that it obeys the investigated dual identities. The relation between Rieamnn and Caputo fractional differences is investigated and the delta and nabla discrete Mittag-Leffler functions are confirmed by solving Caputo type linear fractional difference equations. A nabla integration by parts formula is obtained for Caputo fractional differences as well.