Classical Summation Theorems Research Articles

Applications of Extended Kummer’s Summation Theorem

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 F12; those of Watson, Dixon, Whipple, and Saalschutz for the series F23; 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.

A class of definite integrals involving generalized hypergeometric functions

Recently Masjed-Jamei and Koepf obtained generalizations of various classical summation theorems for the 2F1, 3F2, 4F3, 5F4 and 6F5 generalized hypergeometric series. We evaluate a new class of integrals involving generalized hypergeometric function by employing the results given by Masjed- Jamei and Koepf and MacRobert integral, and we give several special cases.

Summations and transformations for very well-poised hypergeometric functions 2q+5F2q+4(1) and 2q+7F2q+6(1) with arbitrary integral parameter differences

The present paper aims to derive summation and transformation formulae for the generalized very well-poised hypergeometric functions 2q+5F2q+4(1) and 2q+7F2q+6(1) having arbitrary integral parameter differences. These results are derived with the help of Bailey’s transform and the extension of Saalschutz summation theorem for the series ¨ r+3Fr+2(1), where r pairs of parameters differ by positive integers. The particularizations of these generalized identities give classical summation theorems due to Dougall, transformation formula due to Whipple and, other related results. Furthermore, the application of 2q+5F2q+4(1) summation to the limiting case, when q → 1, of one of the Andrews’ q-identities gives a Srivastava-Daoust type multiple hypergeometric series.

Generalized Summation Formulas for the Kampé de Fériet Function

By employing two well-known Euler’s transformations for the hypergeometric function 2F1, 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 2F1 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 2F1 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.

A Note on a Further Extension of Gauss’s Second Summation Theorem with an Application to the Extension of Two Well-Known Combinatorial Identities

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.

A Study of Extensions of Classical Summation Theorems for the Series _{3}F_{2} and _{4}F_{3} with Applications

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.

EVALUATION OF A NEW CLASS OF EULERIAN'S TYPE INTEGRALS INVOLVING GENERALIZED HYPERGEOMETRIC FUNCTIONS

Very recently Masjed-Jamei and Koepf [Axioms 2018, 7 (2), 38] established interesting and useful generalizations of various classical summation theorems for the${}_{2}F_{1}$, ${}_{3}F_{2}$, ${}_{4}F_{3}$, ${}_{5}F_{4}$ and ${}_{6}F_{5}$ 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.

On a New Class of Laplace-Type Integrals Involving Generalized Hypergeometric Functions

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.

Certain Laplace transforms of convolution type integrals involving product of two special pFp functions

Abstract Recently the authors obtained several Laplace transforms of convolution type integrals involving Kummer’s function 1F1 [Appl. Anal. Discrete Math., 2018, 12(1), 257-272]. In this paper, the authors aim at presenting several new and interesting Laplace transforms of convolution type integrals involving product of two special generalized hypergeometric functions pFp by employing classical summation theorems for the series 2F1, 3F2, 4F3 and 5F4 available in the literature.

Some Summation Theorems for Generalized Hypergeometric Functions

Essentially, whenever a generalized hypergeometric series can be summed in terms of gamma functions, the result will be important as only a few such summation theorems are available in the literature. In this paper, we apply two identities of generalized hypergeometric series in order to extend some classical summation theorems of hypergeometric functions such as Gauss, Kummer, Dixon, Watson, Whipple, Pfaff–Saalschütz and Dougall formulas and also obtain some new summation theorems in the sequel.

Some applications of a hypergeometric identity

In this paper, we use a general identity for generalized hypergeometric series to obtain some new applications. The first application is a hypergeometric-type decomposition formula for elementary special functions and the second one is a generalization of the well-known Euler identity \(e^{i\,x} = \cos x + i\,\sin x\) and an extension of hyperbolic functions in the sequel. Applying the mentioned identity on classical hypergeometric orthogonal polynomials and deriving summation formulae for some classical summation theorems are two further applications of this identity.

π AND OTHER FORMULAE IMPLIED BY HYPERGEOMETRIC SUMMATION THEOREMS

By employing certain extended classical summation theorems, several surprising and other formulae are displayed.

NOTE ON THE CLASSICAL WATSON'S THEOREM FOR THE SERIES3F2

Summation theorems for hypergeometric series <TEX>$_2F_1$</TEX> and generalized hypergeometric series <TEX>$_pF_q$</TEX> play important roles in themselves and their diverse applications. Some summation theorems for <TEX>$_2F_1$</TEX> and <TEX>$_pF_q$</TEX> have been established in several or many ways. Here we give a proof of Watson's classical summation theorem for the series <TEX>$_3F_2$</TEX>(1) by following the same lines used by Rakha [7] except for the last step in which we applied an integral formula introduced by Choi et al. [3].

Extension of some classical summation theorems for the generalized hypergeometric series with integral parameter differences

Extension of some classical summation theorems for the generalized hypergeometric series with integral parameter differences

ON REDUCIBILITIES OF CERTAIN TRIPLE HYPERGEOMETRIC SERIES HA AND HC OF SRIVASTAVA

The aim of this research paper is to consider some very interesting reducible cases of HA and HC introduced by Srivastava who actually noticed the existence of three additional complete triple hypergeometric functions HA , HB and HC of the second order in the course of an extensive investigation of Lauricella’s fourteen hypergeometric functions of three variables. The results are derived with the help of known classical summation theorems recently added in the literature due to Lavoie et al. Several results obtained earlier by Kim et al. follow special cases of our main findings.

On a new class of double integrals involving hypergeometric function

The aim of this research paper is to provide a large number of double integrals in the form of four master formulas. A large number of very interesting integrals follow special cases of our main findings. These results are obtained with the help of contiguous extensions of classical summation theorems such as of Watson, Dixon and Whipple on the sum of a $_3F_{2}$. The double integrals established in this paper are simple, interesting, easily established and may be useful.

Generalizations of classical summation theorems for the series 2 F 1 and 3 F 2 with applications

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.

A new hypergeometric transformation of the Rathie–Rakha type

A general transformation involving generalized hypergeometric functions has been recently found by Rathie and Rakha using simple arguments and exploiting Gauss’s summation theorem. In this sequel to the work of Rathie and Rakha, a new hypergeometric transformation formula is derived by their method and by appealing to Gauss’s second summation theorem. In addition, it is shown that the method fails to give similar hypergeometric transformations in the cases of the classical summation theorems of Kummer, Bailey, Watson and Dixon.

Extensions of Certain Classical Summation Theorems for the Series 2 F 1, 3 F 2, and 4 F 3 with Applications in Ramanujan′s Summations

Motivated by the extension of classical Gauss′s summation theorem for the series 2 F 1 given in the literature, the authors aim at presenting the extensions of various other classical summation theorems such as those of Kummer, Gauss′s second, and Bailey for the series 2 F 1, Watson, Dixon and Whipple for the series 3 F 2, and a few other hypergeometric identities for the series 3 F 2 and 4 F 3. As applications, certain very interesting summations due to Ramanujan have been generalized. The results derived in this paper are simple, interesting, easily established, and may be useful.

Dixon’s [formula omitted]-series and identities involving harmonic numbers and the Riemann zeta function

Dixon’s classical summation theorem on F 2 3 ( 1 ) -series is reformulated as an equation of formal power series in an appropriate variable x . Then by extracting the coefficients of x m , we establish a general formula involving harmonic numbers and the Riemann zeta function. Several interesting identities are exemplified as consequences, including one of the hardest challenging identities conjectured by Weideman (2003).