Generalized Extension of Watson's Theorem for the Series $_{3}F_{2}(1)$

Authors

DOI:

https://doi.org/10.29020/nybg.ejpam.v18i3.5760

Keywords:

hypergeometric function, WATSON’S THEOREM

Abstract

The \( _3F_2 \) hypergeometric function holds a pivotal position in the realm of hypergeometric and generalized hypergeometric series. Its significance extends beyond mathematics, impacting various fields such as physics and statistics. 

This research paper aspires to uncover the explicit expression of the \( _3F_2 \) Watson’s classical summation theorem, an endeavor that promises to deepen our understanding and expand the applications of this remarkable function: 

\[
_3F_2\left[a, b, c; \frac{a + b + i + 1}{2}, 2c + j; 1\right].
\]

For any arbitrary \( i \) and \( j \), setting \( i = j = 0 \) leads directly to Watson’s theorem for the series \( _3F_2(1) \). This highlights the theorem’s critical relevance.

Author Biographies

  • Mohamed Awad, Prince Sattam Bin Abdulaziz University

    1- Department of Mathematics, College of Sciences and Humanities in Al-Kharj, Prince
    Sattam Bin Abdulaziz University, Saudi Arabia.

    2- Department of Mathematics, Faculty of Science, Suez Canal University, EGYPT.

  • Medhat Rakha, Suez Canal University

    Department of Mathematics, Faculty of Science, Suez Canal University, EGYPT.

  • Asmaa Mohammed, Suez Canal University

    Department of Mathematics, Faculty of Science, Suez Canal University, EGYPT.

Downloads

Published

2025-08-01

Issue

Vol. 18 No. 3: (July 2025)

Section

Mathematical Analysis

License

Copyright (c) 2025 Mohamed Awad, Medhat Rakha, Asmaa Mohammed

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Upon acceptance of an article by the European Journal of Pure and Applied Mathematics, the author(s) retain the copyright to the article. However, by submitting your work, you agree that the article will be published under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0). This license allows others to copy, distribute, and adapt your work, provided proper attribution is given to the original author(s) and source. However, the work cannot be used for commercial purposes.

By agreeing to this statement, you acknowledge that:

  • You retain full copyright over your work.
  • The European Journal of Pure and Applied Mathematics will publish your work under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0).
  • This license allows others to use and share your work for non-commercial purposes, provided they give appropriate credit to the original author(s) and source.

How to Cite

Generalized Extension of Watson’s Theorem for the Series $_{3}F_{2}(1)$. (2025). European Journal of Pure and Applied Mathematics, 18(3), 5760. https://doi.org/10.29020/nybg.ejpam.v18i3.5760