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An Algebraic Extension of the General Leibniz Product rule for Fractional Indices and Applications

Ryan Wilis

Abstract

This paper presents a reformulation of the Leibniz product rule as a finite sum that expresses the fractional derivative of the product of two differentiable functions. This paper then proves the cases for when the product consists of an arbitrary differentiable function and a Sheffer sequence , and the case for when the product consists of an arbitrary differentiable function and a confluent hypergeometric limiting function with a positive integer order. Finally, this paper provides example expressions for certain Sheffer sequences: any Apell sequence, the falling factorial, the rising factorial, the exponential polynomial, and the Associated Laguerre polynomial.

An Algebraic Extension of the General Leibniz Product rule for Fractional Indices and Applications

Abstract

This paper presents a reformulation of the Leibniz product rule as a finite sum that expresses the fractional derivative of the product of two differentiable functions. This paper then proves the cases for when the product consists of an arbitrary differentiable function and a Sheffer sequence , and the case for when the product consists of an arbitrary differentiable function and a confluent hypergeometric limiting function with a positive integer order. Finally, this paper provides example expressions for certain Sheffer sequences: any Apell sequence, the falling factorial, the rising factorial, the exponential polynomial, and the Associated Laguerre polynomial.
Paper Structure (13 sections, 17 theorems, 28 equations)

This paper contains 13 sections, 17 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. The Truncated Leibniz product rule for non-integers
  3. The Fractional Leibniz product rule of an arbitrary differentiable function and a polynomial with a positive integer degree
  4. The Fractional Leibniz product rule of an arbitrary differentiable function and the confluent hypergeometric limit function with a positive integer order
  5. An Umbral extension of the Truncated General Leibniz Product rule
  6. Applications of the Umbral extension of the General Leibniz Product rule for Some Sheffer Sequences
  7. The Fractional derivative of the product of an Apell sequence and an arbitrary differentiable function
  8. The Fractional derivative of product of the Falling factorial and an arbitrary differentiable function
  9. A Generalization for the iterated logarithm
  10. The Fractional derivative of product of the Rising factorial and an arbitrary differentiable function
  11. The Fractional derivative of the product of the Exponential polynomials and an arbitrary differentiable function
  12. The Fractional derivative of the product of the Associated Laguerre polynomials and any arbitrary differentiable function
  13. Discussion

Key Result

Theorem 2.1

For $n \in \mathbb{N}$

Theorems & Definitions (17)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1
  • Corollary 3.2
  • Theorem 4.1
  • Corollary 4.2
  • Theorem 4.3
  • Lemma 4.4
  • Corollary 4.5
  • ...and 7 more