Chain rule formula and generalized mean value theorem for nabla fractional differentiation on time scale

TL;DR

This paper introduces a chain rule formula and a generalized mean value theorem for nabla fractional derivatives on time scales, unifying continuous and discrete analysis and enabling new applications like summing finite series.

Contribution

It proposes a modified nabla fractional derivative definition and develops new fundamental theorems for it on time scales, advancing fractional calculus theory.

Findings

Established a chain rule formula for nabla fractional derivatives.

Proved a generalized mean value theorem for nabla fractional differentiation.

Applied results to compute sums of finite series.

Abstract

The nabla fractional derivative, which was introduced by Gogoi et.al., generalized the ordinary derivative with non-integer order, and unifies the continuous and discrete analysis using backward operator. In this study, we proposed a modification of their definition. The main focus of this work is to introduce a chain rule formula and a generalized mean value theorem for nabla fractional differentiation on time scale. Results of this study will be applied in finding the sum of a finite series.