On discrete generalized nabla fractional sums and differences

TL;DR

This paper introduces a new class of discrete nabla fractional operators, defining generalized sums and differences of Riemann-Liouville and Caputo types, and explores their properties and relationships with existing operators.

Contribution

It proposes a novel framework for discrete nabla fractional sums and differences, establishing relationships with previous operators and proving fundamental calculus theorems.

Findings

Established relationships between generalized nabla and delta fractional operators

Proved the fundamental theorem of calculus for these operators

Defined Atangana-Baleanu-like fractional operators in the discrete setting

Abstract

This article investigates a class of discrete nabla fractional operators by using the discrete nabla convolution theorem. Inspired by this, we define the discrete generalized nabla fractional sum and differences of Riemann-Liouville and Caputo types. In the process, we give a relationship between the generalized discrete delta fractional operators introduced by Ferreira \cite{Ferreira} and the proposed discrete generalized nabla fractional operators via the dual identities. Also, we present some test examples to justify the relationship. Moreover, we prove the fundamental theorem of calculus for the defined discrete generalized nabla fractional operators. Inspired by the above operators, we define discrete generalized nabla Atangana-Baleanu-like (or Caputo-Fabrizio-like) fractional sum and differences at the end of the article.