# Generalized Extension of Watson's theorem for the series $_{3}F_{2}(1)$

## TL;DR

This paper extends Watson's theorem for the hypergeometric series $_{3}F_{2}(1)$ by deriving an explicit summation formula involving arbitrary parameters, broadening its applicability in mathematics, physics, and statistics.

## Contribution

It provides a generalized explicit expression for Watson's $_{3}F_{2}$ summation theorem with arbitrary parameters, extending the classical result.

## Key findings

- Derived a new summation formula for $_{3}F_{2}$ with arbitrary parameters
- Generalized Watson's theorem to include additional parameters
- Enhanced the applicability of $_{3}F_{2}$ series in various fields

## Abstract

The $_{3}F_{2}$ hypergeometric function plays a very significant role in the theory of hypergeometric and generalized hypergeometric series. Despite that $_{3}F_{2}$ hypergeometric function has several applications in mathematics, also it has a lot of applications in physics and statistics. The fundamental purpose of this research paper is to find out the explicit expression of the $_{3}F_{2}$ Watson's classical summation theorem of the form: \[ _{3}F_{2}\left[ \begin{array} [c]{ccccc}% a, & b, & c & & \\ & & & ; & 1\\ \frac{1}{2}(a+b+i+1), & 2c+j & & & \end{array} \right] \] with arbitrary $i$ and $j$, where for $i=j=0$, we get the well known Watson's theorem for the series $_{3}F_{2}(1)$.