A New Derivation of Extended Watson Summation Theorem Due to Kim et al. with an Application

Confluent representations of hypergeometric functions in one and two variables are firmly established across a range of fields, including applied mathematics, statistics, operations research, physics, and engineering mathematics. Their broad applicability is indisputable. In this article, we will derive the expanded Watson summation theorem for the series 4F3, as introduced by Kim et al., using a novel approach. Additionally, we will evaluate four compelling integrals that involve the generalized hypergeometric function. This note will conclude with a discussion of several specific cases, clearly highlighting the natural emergence of symmetry in the results.