Triple Hypergeometric Functions Research Articles

M-Srivastava hypergeometric functions: integral representations and solution of fractional differential equations

In this paper, we introduce a new extensions of Srivastava’s triple hypergeometric functions HA, HB and HC, by using an modified beta function, which given with a generalized M-series in its kernel. We also introduce a new extension of Appell’s hypergeometric function of the first kind by using the same modified beta function. Furthermore, we give some integral representations of the new extensions of Srivastava’s triple hypergeometric functions. Finally, we obtain solutions of fractional differential equations involving new extensions of Srivastava’s triple hypergeometric functions.

Extended Exton’s Triple and Horn’s Double Hypergeometric Functions and Associated Bounding Inequalities

This paper introduces extensions H4,p and X8,p of Horn’s double hypergeometric function H4 and Exton’s triple hypergeometric function X8, taking into account recent extensions of Euler’s beta function, hypergeometric function, and confluent hypergeometric function. Among the numerous extended hypergeometric functions, the primary rationale for choosing H4 and X8 is their comparable extension type. Next, we present various integral representations of Euler and Laplace types, Mellin and inverse Mellin transforms, Laguerre polynomial representations, transformation formulae, and a recurrence relation for the extended functions. In particular, we provide a generating function for X8,p and several bounding inequalities for H4,p and X8,p. We explore the utilization of the H4,p function within a probability distribution. Most special functions, such as the generalized hypergeometric function, the Beta function, and the p-extended Beta integral, exhibit natural symmetry.

Applications of q-Real Numbers to Triple q-Hypergeometric Functions and q-Horn Functions

The purpose of this article is to study how q-real numbers can be used for computations of convergence regions, q-integral representations of certain multiple triple q-Lauricella functions. The corresponding q-difference equations are also given without proof. In the process, we slightly improve Exton’s original formulas. We also survey the current attempts to generalize the above functions to triple and quadruple hypergeometric functions. Finally, we compute some q-analogues of transformation formulas for Horn functions.

Certain integral transforms involving Appell and Bessel functions and their applications

Abstract This paper deals with the evaluation of certain integral transforms involving the product of certain Appell and Bessel functions with a weight e - γ ⁢ t 2 {e^{-\gamma t^{2}}} . The transformations of these integrals are evaluated in terms of the Appell, Kampé de Fériet and the triple hypergeometric functions. As an application, we studied propagation of generalized Humbert–Gaussian beams (GHGBs) and hypergeometric-Gaussian beams (HyGGBs) in turbulent atmosphere and through an ABCD paraxial optical system. The evaluation of these integral transforms has initiated a great interest in mathematical physics and its applications to laser physics and linear or non-linear optics.

Some Properties of Generalized Srivastava’s Triple Hypergeometric Function $$H_{C,p,q}(\cdot )$$

Some Properties of Generalized Srivastava’s Triple Hypergeometric Function $$H_{C,p,q}(\cdot )$$

On certain results related to the hypergeometric function FK

Motivated by some recent applications of a certain triple series hypergeometric function in the areas of physics and statistics, we investigate further Saran's work (e.g., [Acta Math. 93 (1955), 293–312]) related to this useful hypergeometric function FK. Its analytic continuation and asymptotics are obtained by establishing a new single contour integral representation. With the help of the fractional integration by parts, we prove a new integral identity and discuss in detail its various special cases and their relationship with Koschmieder's work [Acta Math. 79 (1947), 241–254]. Several consequences and special cases of the results presented in this paper are also mentioned.

On Decomposition Formulas Related to the Gaussian Hypergeometric Functions in Three Variables

In this paper, by using certain inverse pairs of symbolic operators introduced by Choi and Hasanov in 2011, we establish several decomposition formulas associated with the Gaussian triple hypergeometric functions. Some transformation formulas for these functions have also been obtained.

On Various Formulas with q-integralsand Their Applications to q-hypergeometric Functions

We present three q-Taylor formulas with q-integral remainder. The two last proofsrequire a slight rearrangement by a well-known formula. The first formula has been given in different form by Annaby and Mansour. We give concise proofs for q-analogues of Eulerian integral formulas for general q-hypergeometric functions corresponding to Erd ́elyi, and for two of Srivastavas triple hypergeometric functions and other functions. All proofs are made in a similar style by using q-integration. We find some new formulas for fractional q-integrals including a series expansion. In the same way, the operator formulas by Srivastava and Manocha find a natural generalization.

An Integral Transform Involving the Product Of Bessel Functions and Whittaker Function and Its Application

In this paper, we aim to establish a closed form for an integral transform involving the product of two Bessel functions of the first kind and the Whittaker function. This integral transform is evaluated in terms of the Lauricella’s triple hypergeometric function, which reduces to the generalized Kampe de Feriet function and is interesting to generate some new laser beams and to study the propagation of these waves through free space and atmospheric and maritime turbulent. Besides, some particular cases are derived by using the relations between the Whittaker function and other special functions and orthogonal polynomials. A novel beams family, called “ Lauricella beams ” is introduced and the propagation of Laguerre–Bessel–Gaussian beam through an ABCD optical system is studied.

A Note on Natural Transformation of Exton’s Triple Series Hypergeometric Function

Triple series hypergeometric functions are very important from the applications point of view. Exton had defined 20 triple hypergeometric functions namely X1 , X2 , … , X20. Integral transform technique is widely using for research purpose. Natural transform is a new kind of integral transform and generalization of Laplace transform. In the present research note, we give the Natural integrals of the triple series hypergeometric function due to Exton.

A new extension of Srivastava’s triple hypergeometric functions and their associated properties

AbstractIn this paper, we define a new extension of Srivastava’s triple hypergeometric functions by using a new extension of Pochhammer’s symbol that was recently proposed by Srivastava, Rahman and Nisar [H. M. Srivastava, G. Rahman and K. S. Nisar, Some extensions of the Pochhammer symbol and the associated hypergeometric functions, Iran. J. Sci. Technol. Trans. A Sci. 43 2019, 5, 2601–2606]. We present their certain basic properties such as integral representations, derivative formulas, and recurrence relations. Also, certain new special cases have been identified and some known results are recovered from main results.

Horn's hypergeometric functions with three variables

The work is devoted to obtaining explicit formulas for analytic continuation of quite general hypergeometric series depending on three variables and belonging to the Horn class. We have derived such continuation formulas with respect to one variable for an arbitrary series belonging to the class under consideration. In addition, on the base of several examples we have demonstrated the use of the obtained formulas for construction of analytic continuation with respect to all three variables. The obtained continuation formulas give representations for a triple hypergeometric series outside the domain of its convergence in the form of a combination of other hypergeometric series that also belong to the Horn class. These series satisfy to the same system of partial differential equations as the initial series. The results of the paper are useful for qualitative analysis and calculation of triple hypergeometric functions.

Some results on Srivastava’s triple hypergeometric matrix functions

Inspired by the recent work by Abd-Elmageed et al., who established recursion formulas satisfied by the first Appell matrix function, namely [Formula: see text], Sahai et al. presented various recursion formulas for the Gauss hypergeometric matrix function and all four Appell matrix functions. In this paper, we obtain recursion formulas and infinte summation formulas for Srivastava’s triple hypergeometric matrix functions [Formula: see text], [Formula: see text] and [Formula: see text].

On the incomplete Srivastava’s triple hypergeometric matrix functions

The paper proposes to introduce incomplete Srivastava’s triple hypergeometric matrix functions through application of the incomplete Pochhammer matrix symbols. We also derive certain properties such as matrix differential equation, integral formula, reduction formula, recursion formula, recurrence relation and differentiation formula of the incomplete Srivastava’s triple hypergeometric matrix functions.

On a New Class of Unified Reduction Formulas for Srivastava’s General Triple Hypergeometric Function F(3) [x, y, z

On a New Class of Unified Reduction Formulas for Srivastava’s General Triple Hypergeometric Function F(3) [x, y, z

On the integral representations of some of the Horn's double and Srivastava's triple hypergeometric functions of matrix arguments

We propose to define the Horn's double hypergeometric functions H3 and H4 of matrix arguments and deduce some integral representations for these two functions. Utilizing the first author's definitions (Upadhyaya, Lalit Mohan and Dhami, H.S., Matrix generalizations of multiple hypergeometric functions; #1818, Nov.2001, IMA Preprint Series, University of Minnesota, Minneapolis, U.S.A. (Retrieved from the University of Minnesota Digital Conservancy, http://hdl.handle.net/11299/3706); Upadhyaya, Lalit Mohan, Matrix Generalizations of Multiple Hypergeometric Functions by Using Mathai's Matrix Transform Techniques (Ph.D. Thesis, Kumaun University, Nainital, Uttarakhand, India), #1943, Nov. 2003, IMA Preprint Series, University of Minnesota, Minneapolis, U.S.A. (https://www.ima.umn.edu/sites/default/files/1943.pdfhttp://www.ima.umn.edu/preprints/abstracts/1943ab.pdfhttp://www.ima.umn.edu/preprints/nov2003/1943.pdf http://hdl.handle.net/11299/3955https://zbmath.org/?q=an:1254.33008http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.192.2172\r Choi, Junesang, Hasanov, Anvar and Turaev, Mamasali, Integral representations for Srivastava's hyper-geometric function HA, Honam Mathematical J., Vol. 34, No. 1, (2012), 113–124: http://dx.doi.org/10.5831/HMJ.2012.34.1.113; Choi, Junesang, Hasanov, Anvar and Turaev, Mamasali, Decomposition formulas and integral representations for some Exton hypergeometric functions, Journal of the Chungcheong Mathematical Society., Vol. 24, No. 4 (December 2011), (2011), 745–758) for these two of the Horn's double and the Sri-vastava's triple hypergeometric functions. For proving our results for these functions of matrix arguments we invoke the Mathai's matrix transform technique for real symmetric positive definite matrices as arguments. We conclude by stating the corresponding parallel results for these Horn's double and the Srivastava's triple hypergeometric functions, when their argument matrices are complex Hermitian positive definite, with the remark that these parallel results can be easily proved by following our given lines of proofs and by employing the corresponding known results available in the literature.

On certain q-hypergeometric matrix functions of several variables and their matrix q-difference equations

ABSTRACTWe introduce q-Lauricella matrix functions of three and n-variables. The convergence of these matrix functions is examined for both and cases. The systems of matrix q-difference equations satisfied by these q-Lauricella matrix functions as well as basic Srivastava's triple hypergeometric matrix functions have been determined. Certain integral representations satisfied by these matrix functions have also been established.

Three-loop vacuum integral with four-propagators using hypergeometry * * Supported partly by the National Natural Science Foundation of China (NNSFC) (11535002, 11705045) and the youth top-notch talent support program of the Hebei Province

A hypergeometric function is proposed to calculate the scalar integrals of Feynman diagrams. In this study, we verify the equivalence between the Feynman parametrization and the hypergeometric technique for the scalar integral of the three-loop vacuum diagram with four propagators. The result can be described in terms of generalized hypergeometric functions of triple variables. Based on the triple hypergeometric functions, we establish the systems of homogeneous linear partial differential equations (PDEs) satisfied by the scalar integral of three-loop vacuum diagram with four propagators. The continuation of the scalar integral from its convergent regions to entire kinematic domains can be achieved numerically through homogeneous linear PDEs by applying the element method.

Finite summation formulas of Srivastava’s general triple hypergeometric function

This paper continues the study of finite summation formulas of multivariable hypergeometric functions. Earlier, Wang and Chen [Finite summation formulas of double hypergeometric functions, Int. Transf. Spec. Funct. 28(3) (2016) 239–253] established finite summation formulas of double hypergeometric functions by using some well-known summation theorems. We present here finite summation formulas for Srivastava’s general triple hypergeometric function. Certain particular cases leading to finite summation formulas for Lauricella functions and Srivastava’s triple hypergeometric functions are also presented.

On Generating Functions of Quadruple Hypergeometric Function

In this work, by using Laplace integral representation of quadruple function defined in [1], we introduce new generating functions involving some triple hyper-geometric functions and itself. Some particular cases and consequences of our main results are also considered.