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In 2012, Chu investigated the generalization of classical Watson–Whipple–Dixon summation theorems in the form of analytical formulas. By employing four generalized Watson summation formulas, the objective of this paper is to evaluate a new class of several Eulerian-type integrals (single and double) and Laplace-type integrals involving a hypergeometric function. Several interesting special cases are also given. Symmetry arises spontaneously in the hypergeometric function. [ABSTRACT FROM AUTHOR]

The main purpose of this note is to demonstrate how one can obtain a new and general summation formula for the generalized hypergeometric series 4F³(1). This is achieved by employing two generalizations of the Gauss's second summation theorems obtained earlier by Rakha and Rathie in an identity due to Bailey. Several results obtained earlier by Ali et al. follow special cases of our main findings. [ABSTRACT FROM AUTHOR]

Using a hypergeometric series approach, a general combinatorial identity is found in this note, and among its special cases are well-known and classical combinatorial identities due to Gould and Touchard. [ABSTRACT FROM AUTHOR]

The aim of this note is to provide a natural extension and generalisation of the well-known Gelfond constant eπ using a hypergeometric function approach. An extension is also found for the square root of this constant. Several known mathematical constants are also deduced in hypergeometric form from our newly introduced constant. [ABSTRACT FROM AUTHOR]

In this note, we aim to provide generalizations of (i) Knuth's old sums (or Reed Dawson's identities) and (ii) Riordan's identities, using a hypergeometric series approach. [ABSTRACT FROM AUTHOR]

Exton [Ganita 54 (2003), 13-15] obtained numerous new quadratic transformations involving hypergeometric functions of order two and of higher order by applying various known classical summation theorems to a general transformation formula based on the Bailey transformation. We obtain a generalization of one of the Exton quadratic transformations. The results are derived with the help of a generalization of Dixon's summation theorem for the series 3F2 obtained earlier by Lavoie et al. Several interesting known as well as new special cases and limiting cases are also given. [ABSTRACT FROM AUTHOR]

Very recently Masjed-Jamei and Koepf [Axioms 2018, 7 (2), 38] established interesting and useful generalizations of various classical summation theorems for the 2F1, 3F2, 4F3, 5F4 and 6F5 generalized hypergeometric series. The main aim of this paper is to establish eleven Eulerian's type integrals involving generalized hypergeometric functions by employing these theorems. Several special cases (known and unknown) have also been given. [ABSTRACT FROM AUTHOR]

In the theory of hypergeometric functions of one or several variables, a remarkable amount of mathematicians's concern has been given to develop their transformation formulas and summation identities. Here we aim at generalizing the following transformation formula for the Exton's triple hypergeometric series X12 and X17: (1+2z)-b X17 (a, b, c3; c1, c2, 2 c3; x, y/1+2z, 4z/1+2z) = X12 (a, b; c1, c2, c3 + 1/2; x, y z²). The result are derived with the help of two general hypergeometric identities for the terminating 2F1 (2 ) series which were very recently obtained by Kim et al. Four interesting results closely related to the Exton's transformation formula are also chosen, among ten, to be derived as special illustrative cases of our main findings. The results easily obtained in this paper are simple and (potentially) useful. [ABSTRACT FROM AUTHOR]

The aim of this research paper is to establish a quite general transformation involving the generalized hypergeometric function. Extensions of Kummer's first transformation, Gauss and Kummer summation, and its contiguous results are then applied to obtain a new class of summation formulas involving the generalized hypergeometric 2F2 polynomial, which have not previously appeared in the literature. A few well-known results obtained earlier by Kim et al. (Int. J. Math. Math. Sci. 2010:309503, 2010; Integral Transforms Spec. Funct. 23(6):435-444, 2012) and Exton have been obtained as special cases of our main findings. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we aim to show how one can obtain so far unknown Laplace transforms of three rather general cases of Kummer’s confluent hypergeometric function by employing generalizations of Gauss’s second summation theorem, Bailey’s summation theorem and Kummer’s summation theorem obtained earlier by Lavoie, Grondin and Rathie. The results established may be useful in theoretical physics, engineering and mathematics. [Copyright &y& Elsevier]

By employing two well-known Euler's transformations for the hypergeometric function 2 F 1 , Liu and Wang established numerous general transformation and reduction formulas for the Kampé de Fériet function and deduced many new summation formulas for the Kampé de Fériet function with the aid of classical summation theorems for the 2 F 1 due to Kummer, Gauss and Bailey. Here, by making a fundamental use of the above-mentioned reduction formulas, we aim to establish 32 general summation formulas for the Kampé de Fériet function with the help of generalizations of the above-referred summation formulas for the 2 F 1 due to Kummer, Gauss and Bailey. Relevant connections of some particular cases of our main identities, among numerous ones, with those known formulas are explicitly indicated. [ABSTRACT FROM AUTHOR]

In the theory of generalized hypergeometric functions, classical summation theorems for the series 2 F 1 , 3 F 2 , 4 F 3 , 5 F 4 and 7 F 6 play a key role. Very recently, Masjed-Jamei and Koepf established generalizations of the above-mentioned summation theorems. Inspired by their work, the main objective of the paper is to provide a new class of Laplace-type integrals involving generalized hypergeometric functions p F p for p = 2 , 3 , 4 , 5 and 7 in the most general forms. Several new and known cases have also been obtained as special cases of our main findings. [ABSTRACT FROM AUTHOR]