Very recently, Masjed-Jamei & Koepf [Some summation theorems for generalized hypergeometric functions, Axioms, 2018, 7, 38, 10.3390/axioms 7020038] established some summation theorems for the generalized hypergeometric functions. The aim of this paper is to establish extensions of some of their summation theorems in the most general form. As an application, several Eulerian-type and Laplace-type integrals have also been given. Results earlier obtained by Jun et al. and Koepf et al. follow special cases of our main findings.

By employing generalizations of Gauss's second, Bailey's and Kummer's summation theorems obtained earlier by Rakha and Rathie, we aim to establish unknown Laplace transform of six rather general formulas of generalized hypergeometric function 2F2[a,b;c,d;x]. The results obtained in this paper are simple, interesting, easily established and may be useful in theoretical physics, engineering and mathematics. Results obtained earlier by Kim et al. and Choi and Rathie follow special cases of our main findings.

Recently, Brychkov et al. established several new and interesting reduction formulas for the Humbert functions (the confluent hypergeometric functions of two variables). The primary objective of this study was to provide an alternative and simple approach for proving four reduction formulas for the Humbert function ${\psi }_2$. We construct intriguing series comprising the product of two confluent hypergeometric functions as an application. Numerous intriguing new and previously known outcomes are also achieved as specific instances of our primary discoveries. It is well-known that the hypergeometric functions in one and two variables and their confluent forms occur naturally in a wide variety of problems in applied mathematics, statistics, operations research, physics (theoretical and mathematical) and engineering mathematics, so the results established in this paper may be potentially useful in the above fields. Symmetry arises spontaneously in the abovementioned functions.

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.

The main aim of this research paper is to obtain fifty new class of integrals involving generalized hypergeometric function in the form of two master formulas. One hundred interesting integrals have been obtained as special cases of our main findings. The results are established with the help of generalized Watson's summation theorem obtained earlier by Lavoie, et al. and an interesting integral due to MacRobert.

By employing classical Watson's and Whipple's $_3F_{2}$-summation theorems, recently Liu, et al. have obtained a few Ramanujan type series for $\frac{1}{\pi}$ and deduced twelve interesting formulas for $\frac{1}{\pi}$. The aim of this short research paper is to point out that these twelve interesting formulas for $\frac{1}{\pi}$ can be easily obtained by employing classical Gauss's summation theorem.