Multivariable Hypergeometric Functions Research Articles

Berezin Transform for Solvable Groups

We study the Berezin transform in the context of solvable groups AN (acting on homogeneous cones and Siegel domains) and determine its spectral decomposition, using an explicit integral kernel representation for the associated ‘eigen-operators’ in terms of multivariable hypergeometric functions.

A GENERAL THEOREM FOR THE GENERALIZED WEYL FRACTIONAL INTEGRAL OPERATOR INVOLVING THE MULTIVARIABLE H-FUNCTION

In this paper we establish a very general and useful theorem which interconnects the Laplace transform and the generalized Weyl fractional integral operator involving the multivariable H-function of related functions of several variables. Our main theorem involves a multidimensional series with essentially arbitrary sequence of complex numbers. By suitably assigning different values to these sequences, one can easily evaluate the generalized Weyl fractional integral operator of special functions of several variables. We have illustrated it for Srivastava-Daoust multivariable hypergeometric function. On account of general nature of this function a number of results involving special functions of one or more variables can be obtained merely by specializing the parameters.

Zonal spherical functions on the complex reflection groups and ( n+1, m+1)-hypergeometric functions

The pair of groups, complex reflection group G( r,1, n) and symmetric group S n , is a Gelfand pair. Its zonal spherical functions are expressed in terms of multivariate hypergeometric functions called ( n+1, m+1)-hypergeometric functions. Since the zonal spherical functions have orthogonality, they form discrete orthogonal polynomials. Also shown is a relation between monomial symmetric functions and the ( n+1, m+1)-hypergeometric functions.

Bergman kernel on generalized exceptional hua domain

We have computed the Bergman kernel functions explicitly for two types of generalized exceptional Hua domains, and also studied the asymptotic behavior of the Bergman kernel function of exceptional Hua domain near boundary points, based on Appell's multivariable hypergeometric function

Multivariate hypergeometric functions as τ-functions of Toda lattice and Kadomtsev–Petviashvili equation

We present the q-deformed multivariate hypergeometric functions related to Schur polynomials as τ-functions of the KP and of the two-dimensional Toda lattice hierarchies. The variables of the hypergeometric functions are the higher times of those hierarchies. The discrete Toda lattice variable shifts parameters of hypergeometric functions. The role of additional symmetries in generating hypergeometric τ-functions is explained.

Milne's hypergeometric functions in terms of free fermions

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).

Rational hypergeometric functions

Multivariate hypergeometric functions associated with toric varieties were introduced by Gel'fand, Kapranov and Zelevinsky. Singularities of such functions are discriminants, that is, divisors projectively dual to torus orbit closures. We show that most of these potential denominators never appear in rational hypergeometric functions. We conjecture that the denominator of any rational hypergeometric function is a product of resultants, that is, a product of special discriminants arising from Cayley configurations. This conjecture is proved for toric hypersurfaces and for toric varieties of dimension at most three. Toric residues are applied to show that every toric resultant appears in the denominator of some rational hypergeometric function.

Hypergeometric Solutions of Soliton Equations

We consider multivariable hypergeometric functions related to Schur functions and show that these hypergeometric functions are tau functions of the KP hierarchy and are simultaneously the ratios of Toda lattice tau functions 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 the parameters of the hypergeometric functions. We construct the determinant representation and the integral representation of a special type for the KP tau functions. We write a system of linear differential and difference equations on these tau functions, which play the role of string equations.

Multivariable Al-Salam & Carlitz Polynomials Associated with the TypeA q-Dunkl Kernel

The Al–Salam & Carlitz polynomials are q–generalizations of the classical Hermite polynomials. Multivariable generalizations of these polynomials are introduced via a generating function involving a multivariable hypergeometric function which is the q–analogue of the type–A Dunkl integral kernel. An eigenoperator is established for these polynomials and this is used to prove orthogonality with respect to a certain Jackson integral inner product. This inner product is normalized by deriving a q–analogue of the Mehta integral, and the corresponding normalization of the multivariable Al–Salam & Carlitz polynomials is derived from a Pieri–type formula. Various other special properties of the polynomials are also presented, including their relationship to the shifted Macdonald polynomials and the big–q Jacobi polynomials.

Multivariable hypergeometric functions for ion–atom collisions

In this work we present a correlated wave function for a three-body continuum Coulomb problem. This state is described by the two-variables Φ2 hypergeometric function. We examine the properties of this function and their differences with previous uncorrelated models. The Φ2 wave function can be considered as a final state of ion–atom ionizing collisions, giving rise to both undistorted (Born-Φ2) and distorted (EIS-Φ2) models. We obtain double differential cross sections with the Born-Φ2 theory for proton–helium collisions in the intermediate to high energy regime. They exhibit all the main features of the electronic emission process and agree with the experimental data.

A Continuum Correlated Wave Model for Ion–Atom Collisions

We present analytical calculations of transition matrices in ion-atom ionizing collisions. The final correlated state is represented by a series expansion in terms of two-body Coulombic wave functions. We study the particular case when this series is the multivariable confluent hypergeometric function Φ2. Transition matrices are also represented by a series using undistorted and distorted wave approaches for the initial state. We show that this series is strongly convergent and analyze the contribution of their different terms to the double differential cross sections within the undistorted approach.

Generalized Knizhnik–Zamolodchikov equations and isomonodromy quantization of the equations integrable via the Inverse Scattering Transform: Maxwell–Bloch system with pumping

Canonical quantization of the isomonodromy solutions of equations integrable via the Inverse Scattering Transform leads to generalized Knizhnik–Zamolodchikov equations. One can solve these equations by the off-shell Bethe ansatz method provided the Knizhnik–Zamolodchikov equations are related with the highest weight representations of the corresponding Lie algebras: These solutions can be written in terms of multi-variable generalizations of special functions of the hypergeometric type. In this work, we consider a realization of the above scheme for the Maxwell–Bloch system with pumping: quantum states for this system are found in terms of the multi-variable confluent hypergeometric function.

The Bergman Kernel of Complex Ovals and Multivariable Hypergeometric Functions

We compute explicitly the Bergman and Szegő kernels for a class of pseudoconvex domains. The kernels are expressed in terms of Appell's multivariable hypergeometric functions. These explicit formulas are applied to investigate the asymptotic behavior of the Bergman kernel near some weakly pseudoconvex boundary points.

Transformations of certain generalized Kampé de Fériet functions II

The development of identities of multivariable hypergeometric functions is further extended based upon the methods of the previous study (H. Exton, J. Phys. A29 (1996), 357‐363) these functions occur in various applications in the fields of physics and quantum chemistry.

Analytical and numerical methods for massive two-loop self-energy diagrams

Motivated by the precision results in the electroweak theory studies of two-loop Feynman diagrams are performed. Specifically this paper gives a contribution to the knowledge of massive two-loop self-energy diagrams in arbitrary and especially four dimensions. This is done in three respects: firstly results in terms of generalized, multivariable hypergeometric functions are presented giving explicit series for small and large momenta. Secondly the imaginary parts of these integrals are expressed as complete elliptic integrals. Finally one-dimensional integral representations with elementary functions are derived. They are very well suited for the numerical evaluations.

Calculation of two-loop self-energies in the electroweak Standard Model

Motivated by the results of the electroweak precision experiments, studies of two-loop self-energy Feynman diagrams are performed. An algebraic method for the reduction of all two-loop self-energies to a set of standard scalar integrals is presented. The gauge dependence of the self-energies is discussed and an extension of the pinch technique to the two-loop level is worked out. It is shown to yield a special case of the background-field method which provides a general framework for deriving Green functions with desirable theoretical properties. The massive scalar integrals of self-energy type are expressed in terms of generalized multivariable hypergeometric functions. The imaginary parts of these integrals yield complete elliptic integrals. Finally, one-dimensional integral representations with elementary integrands are derived which are well suited for numerical evaluation.

Some integral representations of multivariable hypergeometric functions

The present paper is devoted to establish two integral representation formulas of multivariable hypergeometric functions. An application of the formulas yields some integral representations of Lauricella hypergeometric seriesFA(n),FB(n),FC(n) andFD(n) inn variables.

Some further reduction formulas for certain classes of-generalized multiple hyper geometric series arising in physical, astrophysical, and quantum chemical applications

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.

Astrophysical spectroscopy and neutron reactions: Integral transforms and Voigt functions

The Voigt functionsK(x, y) andL(x, y) which play an essential role in astrophysical spectroscopy and neutron physics are investigated and generalized from the viewpoint of integral operators. Unified representations and series expansions involving classical functions of mathematical physics and multivariable hypergeometric functions are established. From the delicate asymptotic analysis of Laplace and Hankel integral transforms we extract complete and rigorous asymptotic expansions of the generalized Voigt functions for large values of the variablesx andy which are of great value in the theory of spectral line profiles.

Some Clebsch-Gordan type linearisation relations and other polynomial expansions associated with a class of generalised multiple hypergeometric series arising in physical and quantum chemical applications

In this sequel to Srivastava's work (1987), a number of new expansions (in series of various classes of hypergeometric polynomials) are derived for the general multivariable hypergeometric function which provides an interesting and useful unification of numerous families of hypergeometric functions of one, two or more variables encountered naturally in a wide variety of physical and quantum chemical applications. By suitably specialising some of these polynomial expansions, Clebsch-Gordan type linearisation relations are deduced for the products of several Jacobi or Laguerre polynomials. It is also shown how one of the linearisation relations involving Laguerre polynomials would yield the corrected (and modified) versions of a couple of results given by Niukkanen (1985).