INTEGRAL REPRESENTATIONS FOR SRIVASTAVA'S HYPERGEOMETRIC FUNCTION HB

In 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.

This paper continues the study of recursion formulas of multivariable hypergeometric functions. In [Recursion formulas for multivariable hypergeometric functions, Asian-Eur. J. Math. 8(4) (2015) Article ID: 1550082, 50 pp.], the authors have given the recursion formulas for three-variable Lauricella functions, three Srivastava’s triple hypergeometric functions and four [Formula: see text]-variable Lauricella functions. We present here the recursion formulas for the general triple hypergeometric function.

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.

On the Euler integral representations of hypergeometric functions in several variables

On Boundary Values of Hypergeometric Functions of Several Variables

Fractional integration of the H-function of several variables

In this article , we explore the properties and relations of the hypergeometric superhyberbolic functions in terms of generalized hypergeometric functions. Then we express the hypergeometric superhyberbolic functions with the help of hypergeometric functions to obtain an integral representation related to classical results within the theory of generalized hypergeometric functions. The work emphasises conditions of convergence, and special cases where the results can be expressed in a simple form and provide a unified framework for evaluating integrals involving hypergeometric functions.

In this paper, we obtain solutions of some multiple series identities involving bounded multiple sequences. We also derive hypergeometric forms of these identities involving Kampé de Fériet double hypergeometric function, Srivastava’s triple hypergeometric function.

In the spirit of Hasanov, Srivastava, and Turaev (2006), we introduce new inverse operators together with a more general operator and find a summation formula for the last one. Based on these operators and the earlier known q-analogues of the Burchnall-Chaundy operators, we find 15 symbolic operator formulas. Then, 10 expansions for the q-analogues of Srivastava’s three triple hypergeometric functions in terms of ϕ34q-hypergeometric and q-Kampé de Fériet functions are derived. These expansions readily reduce to 10 new expansions for the three triple Srivastava hypergeometric functions in terms of F34 hypergeometric and Kampé de Fériet functions.

Recently, Opps, Saad and Srivastava [Recursion formulas for Appell’s hypergeometric function [Formula: see text] with some applications to radiation field problems, Appl. Math. Comput. 207 (2009) 545–558] presented the recursion formulas for Appell’s function [Formula: see text] and also gave its applications to radiation field problems. Then Wang [Recursion formulas for Appell functions, Integral Transforms Spec. Funct. 23(6) (2012) 421–433] obtained the recursion formulas for Appell functions [Formula: see text] and [Formula: see text]. In our investigation here, we derive the recursion formulas for 14 three-variable Lauricella functions, three Srivastava’s triple hypergeometric functions and four [Formula: see text]-variable Lauricella functions.

On certain results related to the hypergeometric function FK

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.

Srivastava noticed the existence of three additional complete triple hypergeometric functions <TEX>$H_A$</TEX>, <TEX>$H_B$</TEX> and <TEX>$H_C$</TEX> of the second order in the course of an extensive investigation of Lauricella's fourteen hypergeometric functions of three variables. In 2004, Rathie and Kim obtained four summation formulas containing a large number of very interesting reducible cases of Srivastava's triple hypergeometric series <TEX>$H_A$</TEX> and <TEX>$H_C$</TEX>. Here we are also aiming at presenting two unified summation formulas (actually, including 62 ones) for some reducible cases of Srivastava's <TEX>$H_C$</TEX> with the help of generalized Dixon's theorem and generalized Whipple's theorem on the sum of a <TEX>$_3F_2$</TEX> obtained earlier by Lavoie et al.. Some special cases of our results are also considered.

We present the fermionic representation for the q-deformed hypergeometric functions related to Schur polynomials. We show that these multivariate hypergeometric functions are tau-functions of the KP hierarchy, and at the same time they are the ratios of Toda lattice tau-functions, considered by Takasaki, evaluated at certain values of higher Toda lattice times. The variables of the hypergeometric functions are related to the higher times of those hierarchies via a Miwa change of variables. The discrete Toda lattice variable shifts parameters of hypergeometric functions. Hypergeometric functions of type pΦs can also be viewed as a group 2-cocycle for the ΨDO on the circle (the group times are higher times of TL hierarchy and the arguments of a hypergeometric function). We obtain the determinant representation and the integral representation of a special type of KP tau-functions, these results generalize some of the results of Milne concerning multivariate hypergeometric functions. We write down a system of partial differential equations for these tau-functions (string equations).

Very recently the authors have obtained a very interesting reduction formula for the Srivastava's triple hypergeometric series (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 (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.