A New Approach to Discrete Integration and Its Implications for Delta Integrable Functions

Mohammed M. Al-Shamiri, V. Rexma Sherine, G. Britto Antony Xavier, D. Saraswathi, T. G. Gerly, P. Chellamani (), Manal Z. M. Abdalla, N. Avinash and M. Abisha
Additional contact information
Mohammed M. Al-Shamiri: Department of Mathematics, Faculty of Science and Arts, King Khalid University, Muhayl Assir 61913, Saudi Arabia
V. Rexma Sherine: Department of Mathematics, Sacred Heart College, Tirupattur 635601, India
G. Britto Antony Xavier: Department of Mathematics, Sacred Heart College, Tirupattur 635601, India
D. Saraswathi: Department of Mathematics, Sacred Heart College, Tirupattur 635601, India
T. G. Gerly: Department of Mathematics, Sacred Heart College, Tirupattur 635601, India
P. Chellamani: Department of Mathematics, St. Joseph’s College of Engineering, OMR, Chennai 600119, India
Manal Z. M. Abdalla: Department of Mathematics, Faculty of Science and Arts, King Khalid University, Muhayl Assir 61913, Saudi Arabia
N. Avinash: Department of Mathematics, Sacred Heart College, Tirupattur 635601, India
M. Abisha: Department of Mathematics, Sacred Heart College, Tirupattur 635601, India

Mathematics, 2023, vol. 11, issue 18, 1-25

Abstract: This research aims to develop discrete fundamental theorems using a new strategy, known as delta integration method, on a class of delta integrable functions. The ν th-fractional sum of a function f has two forms; closed form and summation form. Most authors in the previous literature focused on the summation form rather than developing the closed-form solutions, which is to say that they were more concerned with the summation form. However, finding a solution in a closed form requires less time than in a summation form. This inspires us to develop a new approach, which helps us to find the closed form related to n th-sum for a class of delta integrable functions, that is, functions with both discrete integration and n th-sum. By equating these two forms of delta integrable functions, we arrive at several identities (known as discrete fundamental theorems). Also, by introducing ∞ -order delta integrable functions, the discrete integration related to the ν th-fractional sum of f can be obtained by applying Newton’s formula. In addition, this concept is extended to h -delta integration and its sum. Our findings are validated via numerical examples. This method will be used to accelerate computer-processing speeds in comparison to summation forms. Finally, our findings are analyzed with outcomes provided of diagrams for geometric, polynomial and falling factorial functions.

Keywords: closed form; summation form; Newton’s formula; discrete integration; delta integrable function; fractional sum; value analysis (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/11/18/3872/pdf (application/pdf)
https://www.mdpi.com/2227-7390/11/18/3872/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:11:y:2023:i:18:p:3872-:d:1237473

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().