Recursion formulas for Srivastava’s general triple hypergeometric function

A multivariable hypergeometric function considered recently by Niukkanen and Srivastava (1984, 1985) provides an interesting unification of the generalised hypergeometric function pFq of one variable, Appell and Kampe de Feriet functions of two variables, and Lauricella functions of n variables, and also of many other hypergeometric series which arise naturally in various physical and quantum chemical applications. As pointed out by Srivastava, the multivariable hypergeometric function is an obvious special case of the generalised Lauricella function of n variables, which was first introduced and studied by Srivastava and Daoust (1969). By employing such connections of this multivariable hypergeometric function with the more general multiple hypergeometric functions several interesting and useful properties of this function have been studied, many of which have not been given by Niukkanen. The author derives a number of new reduction formulae for the multivariable hypergeometric function from substantially more general identities involving multiple series with essentially arbitrary terms. Some interesting summation formulae for the multivariable hypergeometric function with x1= . . . =xn=1 and x1= . . . =xn=-1 are also presented.

The multivariable hypergeometric function $$F_{q_0 :q_1 ;...;q_n }^{P_0 :P_1 ;...;P_n } \left( {\begin{array}{*{20}c} {x_1 } \\ \vdots \\ {x_n } \\ \end{array} } \right),$$ considered recently by A. W. Niukkanen and H.M. Srivastava, is known to provide an interesting unification of the generalized hypergeometric functionp F q of one variable, Appell and Kampe de Feriet functions of two variables, and Lauricella functions ofn variables, as also of many other hypergeometric series which arise naturally in various physical, astrophysical, and quantum chemical applications. Indeed, as already pointed out by Srivastava, this multivariable hypergeometric function is an obvious special case of the generalized Lauricella function ofn variables, which was first introduced and studied by Srivastava and M. C. Daoust. By employing such fruitful connections of this multivariable hypergeometric function with much more general multiple hypergeometric functions studied in the literature rather systematically and widely, Srivastava presented several interesting and useful properties of this function, most of which did not appear in the work of Niukkanen. The object of this sequel to Srivastava's work is to derive a further reduction formula for the multivariable hypergeometric function from substantially more general identities involving multiple series with essentially arbitrary terms. Some interesting connections of the results considered here with those given in the literature, and some indication of their applicability, are also provided.

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).

I. M. Gel’fand and his collaborators created the theory of general hypergeometric functions (multivariate hypergeometric functions on lattices and hypergeometric functions on Grassmannians). These functions are multivariate generalizations of hypergeometric functions of one variable. They are solutions of certain systems of differential equations (general hypergeometric systems) and are related to the Radon transform. At the present time, a deep connection of these hypergeometric functions with representations of groups is absent. But it is clear that this connection exists. Remark that the theory of q-analogues of Grassmannians and of related generalizations of Gel’fand hypergeometric functions is under elaboration (see, for example, [432]).

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.

While investigating the Lauricella's list of 14 complete second-order hypergeometric series in three variables, Srivastava noticed the existence of three additional complete triple hypergeometric series of the second order, which were denoted by <TEX>$H_A$</TEX>, <TEX>$H_B$</TEX> and <TEX>$H_C$</TEX>. Each of these three triple hypergeometric functions <TEX>$H_A$</TEX>, <TEX>$H_B$</TEX> and <TEX>$H_C$</TEX> has been investigated extensively in many different ways including, for example, in the problem of finding their integral representations of one kind or the other. Here, in this paper, we aim at presenting further integral representations for the Srivatava's triple hypergeometric function <TEX>$H_B$</TEX>.

We present the Olsson.wl Mathematica package which aims to find transformations for some classes of multivariable hypergeometric functions. It is based on a well-known method developed by P. O. M. Olsson in J. Math. Phys. 5, 420 (1964) to derive the analytic continuations of the Appell F_1 double hypergeometric series using the linear transformations of the Gauss _2F_1 hypergeometric function. We provide a brief description of the method of Olsson and demonstrate the use of the commands of the Olsson.wl package using some examples. In particular, we reproduce various results of the literature on multivariable hypergeometric functions and show practical applications of this package in the derivation of novel formulas. In the context of high energy physics, we also demonstrate how it can be used to disentangle some known results about the analytic continuation of some series representations of the one-loop pentagon in multi-Regge kinematics and D=6-2ϵ. We also present companion package, called ROC2.wl, which is dedicated to the derivation of the regions of convergence of double hypergeometric series. This package can be used independently of Olsson.wl.

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.

We define a new type of multivariable multiple hypergeometric functions in this paper, which is inspired by Exton's multiple hypergeometric functions given by in [13]. Then, for these functions, we obtain some certain type linear generating functions. After that, we derive a variety classes of multilinear and multilateral generating functions for a family of the multivariable multiple hypergeometric functions. In addition, by employing the Erkus-Srivastava polynomials (see [11]) and the fourth type multivariable Horn functions (see [13]), we have also provided some of its conclusions.

In wireless communication channels, the performance of the wireless communication systems deteriorate due to the severe influence of interference, multipath fading, path loss, shadowing, and noise, respectively. In this paper, we investigate the performance of the system transmitting over correlated ź---µ frequency selective fading channels in terms of average error rates. Based on moment generating function approach, closed form expressions for average error probabilities for the system are derived and represented in terms of Appell's hypergeometric functions and Lauricella's multivariate hypergeometric functions. In addition, probability density function based approach is employed to determine the formulae for average error probabilities of the system operating over correlated ź---µ multipath fading channels. The numerical results reveal that, the effects of correlation, decaying power factor and signal constellation size can be reduced using frequency, spatial antennas and path diversities, respectively. Furthermore, we confirm the correctness of the analytical approaches through Monte Carlo simulation technique.

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.

We give a brief account of the key properties of elliptic hypergeometric integrals—a relatively recently discovered top class of transcendental special functions of hypergeometric type. In particular, we describe an elliptic generalization of Euler’s and Selberg’s beta integrals, elliptic analogue of the Euler–Gauss hypergeometric function and some multivariable elliptic hypergeometric functions on root systems. The elliptic Fourier transformation and corresponding integral Bailey lemma technique is outlined together with a connection to the star-triangle relation and Coxeter relations for a permutation group. We review also the interpretation of elliptic hypergeometric integrals as superconformal indices of four dimensional supersymmetric quantum field theories and corresponding applications to Seiberg type dualities.

In this paper, we established new results of the product of a hypergeometric function with the multivariable 𝐴-function by applying definite integrals. Several other new and known results can be obtained from our main theorems.

We give a brief account of the key properties of elliptic hypergeometric integrals -- a relatively recently discovered top class of transcendental special functions of hypergeometric type. In particular, we describe an elliptic generalization of Euler's and Selberg's beta integrals, elliptic analogue of the Euler-Gauss hypergeometric function and some multivariable elliptic hypergeometric functions on root systems. The elliptic Fourier transformation and corresponding integral Bailey lemma technique is outlined together with a connection to the star-triangle relation and Coxeter relations for a permutation group. We review also the interpretation of elliptic hypergeometric integrals as superconformal indices of four dimensional supersymmetric quantum field theories and corresponding applications to Seiberg type dualities.

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.