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In the theory of hypergeometric series and generalized hypergeometric series, classical summation theorems, such as the two of Gauss and those of Kummer and Bailey for the series F 1 2 ; those of Watson, Dixon, Whipple, and Saalschutz for the series F 2 3 ; and others, play a key role. Applications of these classical summation theorems are well known. Berndt pointed out that a large number of interesting summations (including Ramanujan's summations and the Gregory–Leibniz pi summation) can be obtained very quickly by employing the above-mentioned classical summation theorems. Also, several interesting results involving products of generalized hypergeometric series have been obtained by Bailey by employing the above-mentioned classical summation theorems. Recently, the above-mentioned classical summation theorems have been generalized and extended. In our present investigations, our aim is to demonstrate the applications of the extended Kummer's summation theorem in establishing (i) extensions of Gauss's second summation theorem and Bailey's summation theorem; (ii) extensions of several summations (including Ramanujan's summations); (iii) extensions of several results involving products of generalized hypergeometric series; and (iv) an extension of classical Dixon's summation theorem. As special cases, we recover several known summations (including several Ramanujan summations and the Gregory–Leibniz pi summation) and various results involving products of generalized hypergeometric series due to Bailey. [ABSTRACT FROM AUTHOR]

A collection of functions organized according to their indexing based on non-negative integers is grouped by the common factor of fixed integer N. This grouping results in a summation of N series, each consisting of functions partitioned according to this modulo N rule. Notably, when N is equal to two, the functions in the series are divided into two subseries: one containing even-indexed functions and the other containing odd-indexed functions. This partitioning technique is widely utilized in the mathematical literature and finds applications in various contexts, such as in the theory of hypergeometric series. In this paper, we employ this partitioning technique to establish four distinct families of summation formulas for F 3 4 (1) hypergeometric series. Subsequently, we leverage these summation formulas to introduce eight categories of integral formulas. These integrals feature compositions of Beta function-type integrands and F 2 3 (x) hypergeometric functions. Additionally, we highlight that our primary summation formulas can be used to derive some well-known summation results. [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]

In a number of problems in applied mathematics, physics (theoretical and mathematical), statistics, and other fields the hypergeometric functions of one and several variables naturally appear. Hypergeometric functions in one and several variables have several known applications today. The Appell's four functions and the Horn's functions have shown to be particularly useful in providing solutions to a variety of problems in both pure and applied mathematics. The Hubbell rectangular source and its generalization, non-relativistic theory, and the hydrogen dipole matrix elements are only a few examples of the numerous scientific and chemical domains where Appell functions are used. The Appell series is also used in quantum field theory, especially in the evaluation of Feynman integrals. Additionally, since 1985, computational sciences such as artificial intelligence (AI) and information processing (IP) have used the well-known Horn functions as a key idea. In literature, there have been published a significant number of results of double series in particular of Appell and Horn functions in a series of interesting and useful research publications. We find three general transformation formulas between Appell functions F 2 and F 4 and two general transformation formulas between Appell function F 2 and Horn function H 4 in the present study, which are mostly inspired by their work and naturally exhibit symmetry. By using the generalizations of the Kummer second theorem in the integral representation of the Appell function F 2 , this is accomplished. As special cases of our main findings, both previously known and new results have been found. [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 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]

In this short note, we provide a new proof of an interesting and useful reduction formula for the Appell series F3 due to Bailey [On the sum of a terminating 3F2(1), Quart. J. Math. Oxford Ser. [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]

In this note, two interesting integrals involving the product of two generalized hypergeometric functions have been evaluated in terms of gamma function. The results are derived with the help of a known integral involving hypergeometric function recorded in the paper of Lavoie and Trottlier. We also give several interesting special cases of our main findings. [ABSTRACT FROM AUTHOR]

A simple proof is given of a new summation formula recently added in the literature for a terminating r+3Fr+2(1) hypergeometric series for the case when r pairs of numeratorial and denominatorial parameters differ by positive integers. This formula represents an extension of the well-known Saalschütz summation formula for a 3F2(1) series. Two applications of this extended summation formula are discussed. The first application extends two identities given by Ramanujan and the second, which also employs a similar extension of the Vandermonde-Chu summation theorem for the 2F1 series, extends certain reduction formulas for the Kampé de Fériet function of two variables given by Exton and Cvijovi'c & Miller. [ABSTRACT FROM AUTHOR]

Brychkov [Brychkov YuA. Evaluation of some classes of definite and indefinite integrals. Integral Transforms Spec Funct. 2002;13:163–167] evaluated some interesting classes of definite and indefinite integrals involving various special functions and the logarithmic function. Motivated essentially by Brychkov's work [Brychkov YuA. Evaluation of some classes of definite and indefinite integrals. Integral Transforms Spec Funct. 2002;13:163–167], we aim at presenting several classes of definite integrals involving hypergeometric functions and the logarithmic functions, which are expressed explicitly in terms of Psi and generalized zeta functions. Some interesting special cases of our main formulas are given. Relevant connections of the results presented here with those earlier ones are also pointed out. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to establish as many as thirty new closed-form evaluations of the generalized hypergeometric function q+1Fq(z) for q = 2; 3. This is achieved by means of separating the generalized hypergeometric function q+1Fq(z) for q = 1; 2; 3 into even and odd components together with the use of several known infinite series involving reciprocal of the non-central binomial coefficients obtained earlier by L. Zhang and W. Ji. [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]

We obtain two new Thomae-type transformations for hypergeometric series with r pairs of numeratorial and denominatorial parameters differing by positive integers. This is achieved by application of the so-called Beta integral method developed by Krattenthaler and Rao [Symposium on Symmetries in Science (ed. B. Gruber), Kluwer (2004)] to two recently obtained Euler-type transformations. Some special cases are given. [ABSTRACT FROM AUTHOR]

The aim of this research paper is to find explicit expressions of [Equation not available: see fulltext.] and [Equation not available: see fulltext.] each for [InlineEquation not available: see fulltext.]. Known results earlier obtained by Kim et al. and Chu follow special cases of our main findings. The results are derived with the help of generalizations of Gauss second, Kummer and Bailey summation theorems for the series [InlineEquation not available: see fulltext.] obtained earlier by Lavoie et al.MSC: 33C05, 33C20. [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]

The aim of this research paper is to establish fifty new class of integrals involving hypergeometric function in terms of gamma functions. In order to put these fifty integrals, two master formulas have been constructed. The results are derived with the help of an interesting integral due to Lavoie and Trottier and generalized Watson's summation theorem obtained earlier by the Lavoie, Grondhi and Rathi. More than hundred interesting special cases (and four very special results) have also been obtained from our main results. [ABSTRACT FROM AUTHOR]

The aim of this research paper is to establish generalizations of classical Watson's theorem and Fox's theorem by employing a generalized Gauss's second summation theorem obtained earlier by Lavoie, Grondin and Rathie. Several interesting special cases are also given. The results established in this research paper are simple, interesting, easily established and may be useful. [ABSTRACT FROM AUTHOR]

The main objective of this investigation is to derive a generalization of Kummer’s formula: which was proved independently by Ramanujan. The results presented here are obtained with the help of a known generalization of one of Kummer’s summation theorems for 2 F 1 , which was given recently by Lavoie et al. [14]. Several interesting identities closely related to Ramanujan’s formula follow as special cases of our main results. [ABSTRACT FROM AUTHOR]

It is well known that classical summation theorems such as of Gauss, Gauss second, Kummer and Bailey for the series 2 F 1 and Watson, Dixon and Whipple for the series 3 F 2 play an important role in the theory of hypergeometric and generalized hypergeometric series. Applications of these summation theorems are well known. The aim of this research paper is to provide the generalizations of the above-mentioned classical summation theorems in the most general case. [ABSTRACT FROM AUTHOR]

Abstract: Contiguous relations for hypergeometric series contain an enormous amount of hidden information. Applications of contiguous relations range from the evaluation of hypergeometric series to the derivation of summation and transformation formulas for such series. In this paper, a new set of contiguous function relations are established. Applications of such relations to hypergeometric summation formulas and the theory of Jacobi polynomials are presented. [ABSTRACT FROM AUTHOR]

Abstract: By applying various known summation theorems to a general formula based upon Bailey’s transform theorem due to Slater, Exton has obtained numerous new quadratic transformations involving hypergeometric functions of two and of higher order. Some of the results have typographical errors and have been corrected recently by Choi and Rathie. In addition, two new quadratic transformation formulæ were also obtained [Junesang Choi, A.K. Rathie, Quadratic transformations involving hypergeometric functions of two and higher order, EAMJ, East Asian Math. J. 22 (2006) 71–77]. The aim of this research paper is to obtain a generalization of one of the Exton’s quadratic transformation. The results are derived with the help of generalized Kummer’s theorem obtained earlier by Lavoie, Grondin and Rathie. As special cases, we mention six interesting results closely related to that of Exton’s result. [Copyright &y& Elsevier]

In 1979, Apéry proved the irrationality of ζ (2) and ζ (3) . Since then, there has been much research interest in investigating the Apéry-like series for values of Riemann zeta function, Ramanujan-like series for π and other infinite series involving central binomial coefficients. The purpose of this work is to present the first 20 results related to the Apéry-like series in the form of 4 lemmas, each containing 5 results. The Sherman's results are applied to attain this. Thereafter, these 20 results are further used to establish up to 104 results pertaining to the Apéry-like series in the form of 4 theorems, with 26 results each. These findings are finally been described in terms of the generalized hypergeometric functions. Symmetry occurs naturally in the generalized hypergeometric functions. [ABSTRACT FROM AUTHOR]

The generalized hypergeometric functions in one and several variables and their natural generalizations appear in many mathematical problems and their applications. The theory of generalized hypergeometric functions in several variables comes from the fact that the solutions of the partial differential equations appearing in a large number of applied problems of mathematical physics have been expressed in terms of such generalized hypergeometric functions. In particular, the Kampé de Fériet function (in two variables) has proved its practical utility in representing solutions to a wide range of problems in pure and applied mathematics, statistics, and mathematical physics. In this context, in a very recent paper, Progri successfully calculated the 2 F 2 generalized hypergeometric function for a particular set of parameters and expressed the result in terms of the difference between two Kampé de Fériet functions. Inspired by his work, in the present paper, we obtain three results for a terminating 3 F 2 series of arguments 1 and 2, together with a transformation formula of a 3 F 2 (z) generalized hypergeometric function in terms of the difference between two Kampé de Fériet functions. One application of this result is also provided. The paper concludes with six reduction formulas for the Kampé de Fériet function. Of note, symmetry occurs naturally in the generalized hypergeometric functions p F q and the Kampé de Fériet function involving two variables, which are the two most important functions discussed in this paper. [ABSTRACT FROM AUTHOR]

In 1997, Exton [J. Comput. Appl. Math. 88 (1997) 269-274] obtained a general transformation involving hypergeometric functions by elementary manipulation of series. A number of hypergeometric identities that had not been previously recorded in the literature were then deduced by application of Gauss' second summation theorem and other known hypergeometric summation theorems. However, some of the results stated by Exton contain errors. It is the purpose of this note to present the corrected forms of these hypergeometric identities. [ABSTRACT FROM AUTHOR]

By elementary manipulation of series together with summations of Gauss and Saalschütz, Exton deduced a new two-term relation for the hypergeometric function 3F2(1). The aim of this research note is to derive Exton''s result by a very short method. [Copyright &y& Elsevier]

The main objective of this paper is to establish several new closed-form evaluations of the generalized hypergeometric function F q + 1 q z for q = 2 , 3 , 4 , 5. This is achieved by means of separating the generalized hypergeometric function F q + 1 q z (q = 2 , 3 , 4 , 5) into even and odd components together with the use of several known infinite series involving reciprocals of the central binomial coefficients obtained earlier by Lehmer. [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]

The aim of this note is to provide some new Gaussian hypergeometric summation formulae. These are further used to obtain certain new expressions for the product of hypergeometric series. Already obtained results by Bailey, Choi and Rathie and Qureshi et al. follow special cases of our main findings. [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]

The aim of our paper is to present Pδ -transforms of the Kummer's confluent hypergeometric functions by employing the generalized Gauss's second summation the-orem, Bailey's summation theorem and Kummer's summation theorem obtained earlier by Lavoie, Grondin and Rathie [9]. Relevant connections of certain special cases of the main results presented here are also pointed out. [ABSTRACT FROM AUTHOR]

Motivated by recent generalizations of classical theorems for the series 2F1 [Integral Transform. Spec. Funct. 229(11), (2011), 823-840] and interesting Laplace transforms of Kummer's confluent hypergeometric functions obtained by Kim et al. [Math. Comput. Modelling 55 (2012), 1068-1071], first we express generalized summations theorems in explicit forms and then by employing these, we derive various new and useful Laplace transforms of convolution type integrals by using product theorem of the Laplace transforms for a pair of Kummer's confluent hypergeometric function. [ABSTRACT FROM AUTHOR]

New reduction formulas for the Humbert functions (the confluent Appell functions),andare given. [ABSTRACT FROM AUTHOR]

In recent years, various extensions of the well known and useful Kummer's second theorem have been given. In this paper, we aim to give another extension of Kummer's second theorem. [ABSTRACT FROM AUTHOR]

The main objective of this research note is to provide an interesting result for the reducibility of the Kampé de Fériet function. The result is derived with the help of two results for the terminating 3 F2 series very recently obtained by Rakha et al. A few interesting special cases have also been given. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Very recently the authors have obtained a very interesting reduction formula for the Srivastava's triple hypergeometric series F(3)(x, y, z) by applying the so-called Beta integral method to the Henrici's triple product formula for the hypergeometric series. In this sequel, we also present three more interesting reduction formulas for the function F(3)(x, y, z) by using the well known identities due to Bailey and Ramanujan. The results established here are simple, easily derived and (potentially) useful. [ABSTRACT FROM AUTHOR]

Recently, Masjed-Jamei and Koepf established the extension of several classical summation theorems (including Gauss's second summation theorem). Our aim in this paper is to establish a further extension of Gauss's second summation formulas due to Masjed-Jamei and Koepf in the most general form. The result is then applied to obtain extensions of (i) Knuth's old sum (or the Reed Dawson identity) and (ii) Riordan's identity in the most general form. A few interesting results are obtained as special cases of our main findings. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish an extension of Kummer's second theorem in the form ..., where c = (1/(a + 1))(1/2 - a/d)+ a/d(d + 1), d ≠ 0, -1, -2, . . . . For d = 2a, we recover Kummer's second theorem. The result is derived with the help of Kummer's second theorem and its contiguous results available in the literature. As an application, we obtain two general results for the terminating 3F2(2) series. The results derived in this paper are simple, interesting, and easily established and may be useful in physics, engineering, and applied mathematics. [ABSTRACT FROM AUTHOR]

The aim of this research paper is to establish the following summation formula for the Clausen's series 3F2: (Multiple line equation(s) cannot be represented in ASCII text), where f = c(1+a-b)/(a-c), in three different ways. For c = 1/2a, we have (Multiple line equation(s) cannot be represented in ASCII text), which is already available in the literarute. Our formula is then applied to obtain two general results, one is the Euler's transformation for the series 2F2 and another is the Kummer-type first transformation for the series 2F2 established recently by Paris by following a different method. The results obtained generalize the related results by Exton. [ABSTRACT FROM AUTHOR]

In this paper we consider generalizations of the classical Dixon's theorem and the classical Whipple's theorem on the sum of a 3F2. The results are derived with the help of generalized Watson's theorem obtained earlier by Mitra. A large number of results contiguous to Dixon's and Whipple's theorems obtained earlier by Lavoie, Grondin and Rathie, and Lavoie, Grondin, Rathie and Arora follow special cases of our main findings. [ABSTRACT FROM AUTHOR]

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 ψ 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. [ABSTRACT FROM AUTHOR]

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]

The aim of this note is to provide sums of a unified class of series of the form ... For i = 1 and a = n (∈ N), we recover the well known sum of the series due to Vowe and Seiffert. Several other known results due to Srivastava and Kim et al. can be considered as special cases of our result. [ABSTRACT FROM AUTHOR]

In this paper, we establish sixteen interesting generalized fractional integral and derivative formulas including their composition formulas by using certain integral transforms involving generalized (p , q) -Mathieu-type series. [ABSTRACT FROM AUTHOR]