Dual identities in fractional difference calculus within Riemann

TL;DR

This paper explores dual identities in Riemann fractional calculus, linking nabla and delta operators and establishing relations between left and right fractional sums and differences, with solutions for higher order equations.

Contribution

It introduces dual identities connecting nabla and delta fractional sums and differences, and shows the necessity of using both operators for right fractional differences.

Findings

Derived dual identities for Riemann fractional sums and differences.

Established the relation between left and right fractional sums and differences via the Q-operator.

Provided solution representations for higher order Riemann fractional difference equations.

Abstract

We Investigate two types of dual identities for Riemann fractional sums and 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. These dual identities insist that in the definition of right fractional differences we have to use both the nabla and delta operators. The solution representation for higher order Riemann fractional difference equation is obtained as well.