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

Gaddiel L. Dorado, Mark Allien D. Roble

TL;DR

The paper develops a modified framework for nabla fractional differentiation on time scales, addressing real-valuedness issues for certain orders with a carefully defined neighborhood. It then proves a nabla fractional generalized mean value theorem and two-chain-rule formulations, including an explicit integral form, expanding the calculus of compositions on time scales. A key application is a closed-form expression for the sum of finite series in terms of the nabla fractional derivative, enabling efficient summation identities and classical expansions. Collectively, these results extend backward-difference fractional calculus to unified discrete-continuous settings and provide practical tools for series evaluation and numerical analysis.

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.

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

TL;DR

The paper develops a modified framework for nabla fractional differentiation on time scales, addressing real-valuedness issues for certain orders with a carefully defined neighborhood. It then proves a nabla fractional generalized mean value theorem and two-chain-rule formulations, including an explicit integral form, expanding the calculus of compositions on time scales. A key application is a closed-form expression for the sum of finite series in terms of the nabla fractional derivative, enabling efficient summation identities and classical expansions. Collectively, these results extend backward-difference fractional calculus to unified discrete-continuous settings and provide practical tools for series evaluation and numerical analysis.

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.
Paper Structure (6 sections, 19 theorems, 51 equations)

This paper contains 6 sections, 19 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Nabla Fractional Differentiation on Time Scales
  3. A Modification of Nabla Fractional Derivative
  4. Generalized Mean Value Theorem for Nabla Fractional Derivative
  5. Chain Rule for Nabla Fractional Derivative
  6. Sum of a Finite Series

Key Result

Theorem 1

Let $t\in\mathbb T^k$ and let $f:\mathbb T\rightarrow\mathbb R$ be a function. Then the following statements hold:

Theorems & Definitions (27)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Proposition 2: Constant Rule for Nabla Fractional Differentiation
  • Proposition 3: Identity Rule for Nabla Fractional Differentiation
  • Proposition 4: Linearity of Nabla Fractional Differentiation
  • Definition 2
  • Lemma 1
  • Lemma 2
  • Corollary 1
  • ...and 17 more