Shodh Sari-An International Multidisciplinary Journal
Vol-04, Issue-03(Jul-Sep 2025)
An International scholarly/ academic journal, peer-reviewed/ refereed journal, ISSN : 2959-1376
A Study of Generalized Hypergeometric Function and its Applications in Vary Disciplines
Mahala, Mukesh Kumar1 and Sharma, Rahul2
1Department of Mathematics, University of Engineering & Management Jaipur, Jaipur, Rajasthan, India
2Department of Mathematics, Seth Gyaniram Bansidhar Podar College, Nawalgarh, Rajasthan, India
Received on March 05, 2025
Accepted on May 10, 2025
Published on July 01, 2025
Thematic Classification: Generalized Hypergeometric Function, Mathematical Modeling, Applied Mathematics, Computational Functions, Interdisciplinary Applications.
Abstract
Elementary functions, Bessel functions, Legendre functions and many other special functions are included in the large family of mathematical functions known as generalized hypergeometric functions. A power series with coefficients that are rational functions of the index defines them. They are used in many disciplines, such as engineering, statistics and physics, because of their rich mathematical features and adaptability. The beauty and interdependence of mathematical ideas are demonstrated by the Generalized Hypergeometric Function. Researchers and practitioners from a wide range of disciplines find it to be an indispensable tool due to its unifying power, rich analytical features, and broad applications. Numerous unusual functions are included as particular examples of the generalized Hypergeometric function. Legendre polynomials, Bessel functions, the confluent Hypergeometric function, and numerous more noteworthy examples are also included. An order (q+1) linear homogeneous differential equation is satisfied by the generalized hypergeometric function. In many applications, but especially in mathematical physics, this differential equation is essential. It is possible to write the generalized hypergeometric function in terms of contour integrals, which offers different representations and makes it easier to evaluate some integrals. The generalized Hypergeometric function has a wealth of transformation formulas that allow one Hypergeometric function to be transformed into another with distinct parameters. These transformations are quite useful for examining relationships between various special functions and simplifying expressions.
Keywords: Special Function, Generalized Hypergeometric functions, Recurrence Relations
Impact Statement
This study investigates the generalized hypergeometric function, a powerful and unifying mathematical tool that encompasses a wide range of special and elementary functions. By establishing novel representations and transformation identities, the work enhances understanding of the hypergeometric function’s role in solving complex differential equations, particularly those arising in physical models and computational mathematics. The ability to express numerous classical functions—like Legendre, Bessel, and trigonometric functions—as special cases of the generalized hypergeometric function provides a cohesive framework for theoretical and applied investigations. This contributes to both the simplification of mathematical modelling and the development of efficient computational methods.
About Author
Mr. Mukesh Kumar Mahala is a dedicated academic professional with over 9 years of teaching experience in mathematics at the postgraduate level. He is currently serving as an Assistant Professor at Shri Shraddhanath P.G. College, Jhunjhunu. He has previously held similar roles at Shekhawati P.G. College, Sikar, and Shri Narayan P.G. College, Jaipur. He holds an M.Sc. in Mathematics and a B.Ed. and is presently pursuing a PhD from the University of Engineering & Management, Jaipur.
Dr. Rahul Sharma is currently serving as Associate Professor and Head of the Department of Mathematics at the University of Engineering & Management, Jaipur, with over 13 years of teaching and research experience. He holds a Ph.D. in Mathematics from Amity University Rajasthan, specializing in special functions. Dr. Sharma has published extensively in Scopus and ESCI-indexed journals and actively contributes to academic conferences.
Dr. Vidyadhar Sharma’s primary research interests lie in the fields of Applied & Pure Mathematics, with a specific focus on Fractional Calculus and Special Functions. His work includes studies on subclasses of analytic functions, bi-univalent functions, meromorphic functions, and applications of fractional integral operators.
Cite this Article
APA 7th Style
Mahala, M. K., & Sharma, R. (2025). A study of generalized hypergeometric function and its applications in vary disciplines. Shodh Sari-An International Multidisciplinary Journal, 4(03), 46–56. https://doi.org/10.59231/SARI7836.
Chicago 17th Style
Mahala, Mukesh Kumar, and Rahul Sharma. “A Study of Generalized Hypergeometric Function and Its Applications in Vary Disciplines.” Shodh Sari-An International Multidisciplinary Journal 4, no. 3 (2025): 46–56. https://doi.org/10.59231/SARI7836.
MLA 9th Style
Mahala, Mukesh Kumar and Sharma, Rahul. “A Study of Generalized Hypergeometric Function and Its Applications in Vary Disciplines.” Shodh Sari-An International Multidisciplinary Journal, vol. 4, no. 3, 2025, pp. 46-56, https://doi.org/10.59231/SARI7836.
Statements and Declarations
Peer-Review Method: This mathematical study underwent a double-blind peer-review process by two independent external experts in Mathematical Analysis and Applied Mathematics to ensure technical accuracy and theoretical rigor.
Competing Interests: The authors (Mukesh Kumar Mahala and Rahul Sharma) declare that they have no financial or personal conflicts of interest that could have inappropriately influenced the research findings or mathematical derivations presented in this study.
Funding: This research was conducted as part of the authors’ scholarly activities at the University of Engineering & Management Jaipur and Seth Gyaniram Bansidhar Podar College. No specific external grants or commercial funding were received for this work.
Data Availability: The study is based on theoretical mathematical derivations and a review of existing literature on hypergeometric functions. All formulas and analytical data used are contained within the article or available through standard mathematical archives.
Licence: A Study of Generalized Hypergeometric Function and its Applications in Vary Disciplines © 2025 by Mahala, M. K. and Sharma, R. is licensed under CC BY-NC-ND 4.0. This work is published by the International Council for Education Research and Training (ICERT).
Ethics Approval: As this research consists of a theoretical mathematical analysis and does not involve human participants or animal subjects, it was deemed exempt from formal ethical review by the Institutional Research Committees of the authors’ respective affiliations.
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