Generalized Extension of Watson's Theorem for the Series ₃F₂(1)

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: \[_3F_2[a, b, c; a + b + i + 1/2, 2c + j; 1].\] 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.