Fractional differences and sums with binomial coefficients

TL;DR

This paper develops discrete versions of fractional differences and sums using binomial coefficients, showing they align with traditional Riemann fractional calculus through the Q-operator and dual identities.

Contribution

It formulates delta and nabla discrete fractional operators based on the binomial theorem and proves their equivalence to Riemann fractional differences and sums.

Findings

Discrete fractional differences and sums coincide with Riemann fractional operators.

Discrete versions are formulated for both left and right fractional integrals and derivatives.

The approach bridges the binomial theorem method with classical fractional calculus.

Abstract

In fractional calculus there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and derivatives. The second approach is by iterating the derivative and then defining a fractional order by making use of the binomial theorem to obtain Gr\"{u}nwald-Litnikov fractional derivatives. In this article we formulate the delta and nabla discrete versions for left and right fractional integrals and derivatives representing the second approach. Then, we use the discrete version of the Q-operator and some discrete fractional dual identities to prove that the presented fractional differences and sums coincide with the discrete Riemann ones describing the first approach.