Hypergeometric Functions Of Several Variables Research Articles

EXPANSIONS OF KAMP ́E DE F ́ERIET HYPERGEOMETRIC FUNCTIONS

When studying the properties of the hypergeometric functions in two variables, expansion formulas are very important, allowing one to represent a function of two variables in the form of an infinite sum of products of several hypergeometric Gaussian functions, and this in turn facilitates the process of studying the properties of functions in two variables. Burchnall and Chaundy, in 1940--41, using the symbolic method, obtained more than 15 pairs of expansions for the second-order double hypergeometric Appell and Humbert functions. In order to find expansion formulas for functions depending on three or more variables, Hasanov and Srivastava introduced symbolic operators, with the help of which they were able to expand a whole class of hypergeometric functions of several variables. Hasanov, Turaev and Choi defined so-called H-operators that make it possible to find expansions for generalized hypergeometric functions of one variable. In addition, applications of these H-operators to the expansion of the hypergeometric functions of two and three variables of second order are known. On the other hand, thanks to the Kampe de Feriet functions, solutions of the boundary value problems for some degenerate and singular partial differential equations can be written in explicit forms. In this paper, expansion formulae are obtained for the hypergeometric Kampe de Feriet functions of the superior order. Some Kampe de Feriet functions are expanded in terms of the Appell and Humbert functions as illustrative examples

Some New Results for the Kampé de Fériet Function with an Application

The generalized hypergeometric functions in one and several variables and their natural generalizations appear in many mathematical problems and their applications. The theory of generalized hypergeometric functions in several variables comes from the fact that the solutions of the partial differential equations appearing in a large number of applied problems of mathematical physics have been expressed in terms of such generalized hypergeometric functions. In particular, the Kampé de Fériet function (in two variables) has proved its practical utility in representing solutions to a wide range of problems in pure and applied mathematics, statistics, and mathematical physics. In this context, in a very recent paper, Progri successfully calculated the 2F2 generalized hypergeometric function for a particular set of parameters and expressed the result in terms of the difference between two Kampé de Fériet functions. Inspired by his work, in the present paper, we obtain three results for a terminating 3F2 series of arguments 1 and 2, together with a transformation formula of a 3F2(z) generalized hypergeometric function in terms of the difference between two Kampé de Fériet functions. One application of this result is also provided. The paper concludes with six reduction formulas for the Kampé de Fériet function. Of note, symmetry occurs naturally in the generalized hypergeometric functions pFq and the Kampé de Fériet function involving two variables, which are the two most important functions discussed in this paper.

Dirichlet Averages of Generalized Mittag-Leffler Type Function

Since Gösta Magus Mittag-Leffler introduced the so-called Mittag-Leffler function in 1903 and studied its features in five subsequent notes, passing the first half of the 20th century during which the majority of scientists remained almost unaware of the function, the Mittag-Leffler function and its various extensions (referred to as Mittag-Leffler type functions) have been researched and applied to a wide range of problems in physics, biology, chemistry, and engineering. In the context of fractional calculus, Mittag-Leffler type functions have been widely studied. Since Carlson established the notion of Dirichlet average and its different variations, these averages have been explored and used in a variety of fields. This paper aims to investigate the Dirichlet and modified Dirichlet averages of the R-function (an extended Mittag-Leffler type function), which are provided in terms of Riemann-Liouville integrals and hypergeometric functions of several variables. Principal findings in this article are (possibly) applicable. This article concludes by addressing an open problem.

Hypergeometric Functions of Several Variables and Evaluation of Error Probability in Fading Multichannel System

A method is proposed of evaluation of symbol and/or bit error probabilities for coherent diversity receiving of multipositional signal constructions in communication channel with fadings, which are described with the help of classical and generalized models Multiple-Wave with Diffuse Power (MWDP) fading and of additive white Gaussian noise (AWGN). This method uses the hypergeometric functions of several variables.

Application of Hypergeometric Functions of Several Variables in the Mathematical Theory of Communication: Evaluation of Error Probability in Fading Singlechannel System

A method is proposed of evaluation of symbol and/or bit error probabilities for coherent receiving of multipositional signal constructions in communication channel with fadings, which are described with the help of classical and generalized models Multiple-Wave with Diffuse Power (MWDP) fading and of additive white Gaussian noise (AWGN). This method uses the hypergeometric functions of several variables.

Fundamental Solutions of the Generalized Helmholtz Equation with Several Singular Coefficients and Confluent Hypergeometric Functions of Many Variables

An investigation of applied problems related to heat conduction and dynamics, electromagnetic oscillations and aerodynamics, quantum mechanics and potential theory leads to the study of various hypergeometric functions. The great success of the theory of hypergeometric functions in one variable has stimulated the development of corresponding theory in two and more variables. In the theory of hypergeometric functions, an increase in a number of variables will always be accompanied by a complication in the study of the function of several variables. Therefore, the decomposition formulas that allow us to represent the hypergeometric function of several variables through an infinite sum of products of several hypergeometric functions in one variable are very important, and this, in turn, facilitate the process of studying the properties of multidimensional functions. In the literature, hypergeometric functions are divided into two types: complete and confluent. In all respects, confluent hypergeometric functions including the decomposition formulas, have been little studied in comparison with other types of hypergeometric functions, especially when the dimension of the variables exceeds two. In this paper we define a new class of confluent hypergeometric functions of several variables, study their properties and determine the system of hypergeometric equations that these functions satisfy, because all fundamental solutions of the generalized Helmholtz equation with singular coefficients are written out through one new introduced confluent hypergeometric function of several variables. Using the decomposition formulas which are established here the order of the singularity of the found fundamental solutions of the elliptic equation which mentioned above is determined.

Application of Hypergeometric Functions of Two Variables in Wireless Communication Theory

Hypergeometric functions of several variables are widely used in applied problems, in particular, in the mathematical theory of communication in the problems of calculating the error probabilities of receiving signal constructions. Error probabilities can often be expressed through various types of special functions. One of the most important functions of this kind is the Marcum function and its generalization ℋ-function.

КОНФЛЮЭНТНЫЕ ГИПЕРГЕОМЕТРИЧЕСКИЕ ФУНКЦИИ МНОГИХ ПЕРЕМЕННЫХ И ИХ ПРИМЕНЕНИЕ К НАХОЖДЕНИЮ ФУНДАМЕНТАЛЬНЫХ РЕШЕНИЙ ОБОБЩЕННОГО УРАВНЕНИЯ ГЕЛЬМГОЛЬЦА С СИНГУЛЯРНЫМИ КОЭФФИЦИЕНТАМИ

An investigation of applied problems related to heat conduction and dynamics, electromagnetic oscillations and aerodynamics, quantum mechanics and potential theory leads to the study of various hypergeometric functions. The great success of the theory of hypergeometric functions of one variable has stimulated the development of the corresponding theory for functions of two and more variables. In the theory of hypergeometric functions, an increase in the number of variables will always be accompanied by a complication in the study of the function of several variables. Therefore, the decomposition formulas that allow us to represent the hypergeometric function of several variables in terms of an infinite sum of products of several hypergeometric functions in one variable are very important, and this, in turn, facilitates the process of studying the properties of multidimensional functions. Confluent hypergeometric functions in all respects, including the decomposition formulas, have been little studied in comparison with other types of hypergeometric functions, especially when the dimension of the variables exceeds two. In this paper, we define a new class of confluent hypergeometric functions of several variables, study their properties, give integral representations, and establish decomposition formulas. An important application of confluent functions has been found. It turns out that all fundamental solutions of the generalized Helmholtz equation with singular coefficients are written out through one new introduced confluent hypergeometric function of several variables. Using the decomposition formulas, the order of the singularity of the fundamental solutions of the above elliptic equation is determined.

On the analytic complexity of hypergeometric functions

Hypergeometric functions of several variables resemble functions of finite analytic complexity in the sense that the elements of both classes satisfy certain canonical overdetermined systems of partial differential equations. Otherwise these two sets of functions are very different. We investigate the relation between the two classes of functions and compute the analytic complexity of certain bivariate hypergeometric functions.

On a family of symmetric hypergeometric functions of several variables and their Euler type integral representation

This paper is devoted to the family {Gn} of hypergeometric series of any finite number of variables, the coefficients being the square of the multinomial coefficients (ℓ1+⋯+ℓn)!/(ℓ1!…ℓn!), where n∈Z⩾1. All these series belong to the family of the general Appell–Lauricellaʼs series. It is shown that each function Gn can be expressed by an integral involving the previous one, Gn−1. Thus this family can be represented by a multidimensional Euler type integral, what suggests some explicit link with the Gelfand–Kapranov–Zelevinskyʼs theory of A-hypergeometric systems or with the Aomotoʼs theory of hypergeometric functions. The quasi-invariance of each function Gn with regard to the action of a finite number of involutions of C⁎n is also established. Finally, a particular attention is reserved to the study of the functions G2 and G3, each of which is proved to be algebraic or to be expressed by the Legendreʼs elliptic function of the first kind.

INTEGRAL REPRESENTATIONS FOR SRIVASTAVA'S HYPERGEOMETRIC FUNCTION HB

Abstract. While investigating the Lauricella’s list of 14 completesecond-order hypergeometric series in three variables, Srivastavanoticed the existence of three additional complete triple hypergeo-metric series of the second order, which were denoted by H A , H B and H C . Each of these three triple hypergeometric functions H A ,H B and H C has been investigated extensively in many di erentways including, for example, in the problem of nding their inte-gral representations of one kind or the other. Here, in this paper, weaim at presenting further integral representations for the Srivatava’striple hypergeometric function H A . 1. Introduction and PreliminariesIn the theory of hypergeometric functions of several variables, a re-markably large number of triple hypergeometric functions have beenintroduced and investigated. A comprehensive table of 205 distincttriple hypergeometric functions is provided in the work of Srivastavaand Karlsson [15, Chapter 3]. Out of these 205 distinct triple hyperge-ometric functions, Lauricella [8, p. 114] introduced fourteen completetriple hypergeometric functions of the second order. He denoted histriple hypergeometric functions by the symbols F

Integrable elliptic pseudopotentials

We construct integrable pseudopotentials with an arbitrary number of fields in terms of an elliptic generalization of hypergeometric functions in several variables. These pseudopotentials are multiparameter deformations of ones constructed by Krichever in studying the Whitham-averaged solutions of the KP equation and yield new integrable (2+1)-dimensional systems of hydrodynamic type. Moreover, an interesting class of integrable (1+1)-dimensional systems described in terms of solutions of an elliptic generalization of the Gibbons-Tsarev system is related to these pseudopotentials.

The spherical functions and the Central limit theorem on SU(2,2)/ S( U(2)× U(2))

Let G = SU(2,2), K = S( U(2) × U(2)), and for l ∈ Z , let { τ l } l ∈ z be a one-dimensional K-type and let E l be the line bundle over G/K associated to τ 1. It is shown that the τ l-spherical function on G is given by the hypergeometric functions of several variables. By applying this result, a central limit theorem for the space G/K is obtained.

Singularities of hypergeometric functions in several variables

This paper deals with singularities of nonconfluent hypergeometric functions in several complex variables. Typically such a function is a multi-valued analytic function with singularities along an algebraic hypersurface. We describe such hypersurfaces in terms of the amoebas and the Newton polytopes of their defining polynomials. In particular, we show that the amoebas of classical discriminantal hypersurfaces are solid, that is, they possess the minimal number of complement components.

Some multiple hypergeometric transformations and associated reduction formulas

The main object of the present paper is to derive various classes of double-series identities and to show how these general results would apply to yield some (known or new) reduction formulas for the Appell, Kampé de Fériet, and Lauricella hypergeometric functions of several variables. A number of closely-related linear generating functions for the classical Jacobi polynomials are also investigated.

Some multiple hypergeometric transformations and associated reduction formulas

The main object of the present paper is to derive various classes of double-series identities and to show how these general results would apply to yield some (known or new) reduction formulas for the Appell, Kampé de Fériet, and Lauricella hypergeometric functions of several variables. A number of closely-related linear generating functions for the classical Jacobi polynomials are also investigated.

Two New Degenerate Hypergeometric Functions of Several Variables and Integral Equations Containing These Functions

Two New Degenerate Hypergeometric Functions of Several Variables and Integral Equations Containing These Functions

Determination of exact sample sizes in the Bayesian estimation of the difference of two proportions

Using generalized hypergeometric functions in several variables in a Bayesian context, we compute the exact minimum double-sample size (n 1 , n 2 ) required in the Bernoulli sampling of two independent populations, so that the expected length (or the maximum length) of the highest posterior density credible interval of P = P 1 - P 2 is less than a preset quantity, where P 1 and P 2 are two independent proportions. This precise and computer-intensive approach permits the treatment of this Bayesian sample size determination problem under very general hypotheses and also provides a relationship between the minimal values of n 1 and n 2 . Similar results are derived in an applied Bayesian decision theory context, with a quadratic loss function, and the criteria used are now the posterior risk, the Bayes risk and the expected value of sample information.

Determinants of Period Matrices and an Application to Selberg's Multidimensional Beta Integral

In work on critical values of linear functions and hyperplane arrangements, A. Varchenko (Izv. Akad. Nauk SSSR Ser. Mat.53 (1989), 1206–1235; 54 (1990), 146–158) defined certain period matrices whose entries are Euler-type integrals representing hypergeometric functions of several variables and derived remarkable closed-form expressions for the determinants of those matrices. In this article, we present elementary proofs of some of Varchenko's determinant formulas. By the same method, we obtain proofs of variations of Varchenko's determinants. As an application, we deduce new proofs of the multidimensional beta integrals of Selberg and of Aomoto. Further, we obtain a new proof of a determinant formula of A. Varchenko (Funct. Anal. Appl.25 (1999), 304–305) in which the entries are multidimensional Selberg-type integrals.

Less simple contiguous function relations for hypergeometric functions

Several results for contiguous function relations for hypergeometric functions of several variables are obtained. In these recurrences the coefficients are not all constant, but instead contain linear functions of the variables.