Summation Theorem 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.

Hypergeometric series and generalized harmonic numbers

By computing the higher-order derivatives of some well-known hypergeometric summation theorems, we evaluate many different infinite series containing generalized harmonic functions and central binomial coefficients.

Gauss’ Second Theorem for F12(1/2)-Series and Novel Harmonic Series Identities

Two summation theorems concerning the F12(1/2)-series due to Gauss and Bailey will be examined by employing the “coefficient extraction method”. Forty infinite series concerning harmonic numbers and binomial/multinomial coefficients will be evaluated in closed form, including eight conjectured ones made by Z.-W. Sun. The presented comprehensive coverage for the harmonic series of convergence rate “1/2” may serve as a reference source for readers.

Some Properties of Generalized Apostol-Type Frobenius–Euler–Fibonacci Polynomials

In this paper, by using the Golden Calculus, we introduce the generalized Apostol-type Frobenius–Euler–Fibonacci polynomials and numbers; additionally, we obtain various fundamental identities and properties associated with these polynomials and numbers, such as summation theorems, difference equations, derivative properties, recurrence relations, and more. Subsequently, we present summation formulas, Stirling–Fibonacci numbers of the second kind, and relationships for these polynomials and numbers. Finally, we define the new family of the generalized Apostol-type Frobenius–Euler–Fibonacci matrix and obtain some factorizations of this newly established matrix. Using Mathematica, the computational formulae and graphical representation for the mentioned polynomials are obtained.

Some summation theorems and transformations for hypergeometric functions of Kampé de Fériet and Srivastava

Abstract Owing to the remarkable success of the hypergeometric functions of one variable, the authors present a study of some families of hypergeometric functions of two or more variables. These functions include (for example) the Kampé de Fériet-type hypergeometric functions in two variables and Srivastava’s general hypergeometric function in three variables. The main aim of this paper is to provide several (presumably new) transformation and summation formulas for appropriately specified members of each of these families of hypergeometric functions in two and three variables. The methodology and techniques, which are used in this paper, are based upon the evaluation of some definite integrals involving logarithmic functions in terms of Riemann’s zeta function, Catalan’s constant, polylogarithm functions, and so on.

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.

Q-Analogues of $\pi$-Series by Applying Carlitz Inversions to q-Pfaff-Saalschutz Theorem

By applying multiplicate forms of the Carlitz inverse series relations to the $q$-Pfaff-Saalschtz summation theorem, we establish twenty five nonterminating $q$-series identities with several of them serving as $q$-analogues of infinite series expressions for $\pi$ and $1/\pi$, including some typical ones discovered by Ramanujan (1914) and Guillera.

Implementation of summation theorems of Andrews and Gessel-Stanton

Generalized hypergeometric functions and their natural generalizations in one and several variables appear in many mathematical problems and their applications. Solving partial differential equations encountered in many applied problems of mathematics physics is expressed in terms of such generalized hypergeometric functions. In particular, the Srivastava-Daoust double hypergeometric function (S-D function) has proved its practical utility in representing solutions to a wide range of problems in pure and applied mathematics. In this paper, we introduce two general double-series identities involving bounded sequences of arbitrary complex numbers employing the finite summation theorems of Gessel-Stanton and Andrews for terminating 3F2 hypergeometric series with arguments 3/4 and 4/3, respectively. Using these double-series identities, we establish two reduction formulas for the (S-D function) with arguments z, 3z/4 and z, −4z/3 expressed in terms of two generalized hypergeometric function of arguments proportional to z3 and −z3 respectively. All the results mentioned in the paper are verified numerically using Mathematica Program.

Some Terminating K-Balanced Clausen Hypergeometric Summation Theorems

Some Terminating K-Balanced Clausen Hypergeometric Summation Theorems

Infinite Series Concerning Tails of Riemann Zeta Values

Infinite series involving Riemann’s zeta and Dirichlet’s lambda tails, and weighted by three harmonic-like elementary symmetric functions are examined. By means of integral representations of zeta tails together with the telescopic approach, twelve general summation theorems are established that express these series as coefficients of the bivariate beta function Beta(u,v). By further expanding Beta(u,v) into Laurent series in u and v, several explicit summation formulae are shown as consequences.

π -Formulae from dual series of the Dougall theorem

UDC 517.5 By means of the extended Gould–Hsu inverse series relations, we find that the dual relation of Dougall's summation theorem for the well-poised 7 F 6 -series can be used to construct numerous interesting Ramanujan-like infinite-series expressions for π ± 1 and π ± 2 , including an elegant formula for π -2 due to Guillera.

Some zero-balanced terminating hypergeometric series and their applications

Various families of such Special Functions as the hypergeometric functions of one, two and more variables, and their associated summation, transformation and reduction formulas, are potentially useful not only as solutions of ordinary and partial differential equations, but also in the widespread problems in the mathematical, physical, engineering and statistical sciences. The main object of this paper is first to establish four general double-series identities, which involve some suitably-bounded sequences of complex numbers, by using zero-balanced terminating hypergeometric summation theorems for the generalized hypergeometric series r+1Fr(1) (r = 1, 2, 3) in conjunction with the series rearrangement technique. The sum (or difference) of two general double hypergeometric functions of the Kamp? de F?riet type are then obtained in terms of a generalized hypergeometric function under appropriate convergence conditions. A closed form of the following Clausen hypergeometric function: 3F2 (?27z/4(1?z)3) and a reduction formula for the Srivastava-Daoust double hypergeometric function with the arguments (z,?z/4 ) are also derived. Many of the reduction formulas, which are established in this paper, are verified by using the software program, Mathematica. Some potential directions for further researches along the lines of this paper are also indicated.

Some Summation Theorems for Truncated Clausen Series and Applications

Some Summation Theorems for Truncated Clausen Series and Applications

Some Extensions and Generalizations of Kümmer's Third Summation Theorem

The motive of this research paper is to obtain explicit forms of certain extensions and generalizations of Kümmer's third summation theorem, which have not previously appeared in the literature, by using the summation theorem given by Rakha and Rathie (2011). The results derived in this paper are interesting and may be beneficial.

Some Convergent Summation Theorems For Appell’s Function F1 Having Arguments −1, 1/2

In this paper, we obtain some closed forms of hypergeometric summation theorems for Appell’s function of first kind F1 having the arguments −1, 1/2 with suitable convergence conditions, by adjustment of parameters and arguments in generalized form of first, second and third summation theorems of K¨ummer and others.

Certain cubic reduction formulas involving hypergeometric functions

In this paper, we obtain a new general double infinite series identity (in terms of the sum of three infinite series) involving the bounded sequence of arbitrary complex numbers using Saalschutz summation theorem for terminating Clausen series. As application of our double series identity, we establish two cubic reduction formulas for Srivastava-Daoust double hypergeometric functions in terms of generalized hypergeometric function with suitable convergence conditions. By the theory of analytic continuation, our cubic reduction formula is also valid in -211/25≤R(z)≤3/4 when F(z)=0, using Mathematica software.

EVALUATION OF CERTAIN EXOTIC (1)-SERIES

AbstractA class of exotic $_3F_2(1)$ -series is examined by integral representations, which enables the authors to present relatively easier proofs for a few remarkable formulae. By means of the linearization method, these $_3F_2(1)$ -series are further extended with two integer parameters. A general summation theorem is explicitly established for these extended series, and several sample summation identities are highlighted as consequences.

Expansion formulas for multiple basic hypergeometric series over root systems

We extend expansion formulas of Liu given in 2013 to the context of multiple series over root systems. Liu and others have shown the usefulness of these formulas in Special Functions and number-theoretic contexts. We extend Wang and Ma's generalizations of Liu's work which they obtained using q-Lagrange inversion. We use the An and Cn Bailey transformation and other summation theorems due to Gustafson, Milne, Milne and Lilly, and others, from the theory of An, Cn and Dn basic hypergeometric series.

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.

ON CLAUSEN SERIES $_{3}F_{2}[-m, \alpha, \lambda+3; \beta,\lambda;1]$ WITH APPLICATIONS

In this paper, a summation theorem for the Clausen series is derived. Further, a reduction formula is obtained for the Kamp\'{e} de F\'{e}riet double hypergeometric function. Some special cases are given as applications. A generalization of the reduction and linear transformation formulas is also given in the form of the general double series identity.