Lauricella Hypergeometric Function Research Articles

ε-Expansion for nonplanar six-propagators double boxes

We present calculations for a nonplanar double box with four massless, massive external, and internal legs propagators. The results are expressed for arbitrary exponents of propagators and dimension in terms of Lauricella's hypergeometric functions of three variables and hypergeometric-like multiple series. PACS Nos.: 11.15.Bt, 12.38.Bx

A generalization of beta-type distribution involving ω-Lauricella function in several variables

The object of this paper is to define and study a new generalization of beta-type function in the following form: where λ, u, v, a, c, b j (j = 1, Λ, n) ∈ C with c ≠ 0,−1,−2, Λ; ω, Re(u), Re (a + λ), Re (b j + λ) > 0(j = 1, Λ, n), | arg v| < π and is the ω-generalization of the Lauricella hypergeometric function of n-variables. The previous defined function provides extensions of the various type of beta-functions studied earlier by many authors, notably by Ben Nakhi and Kalla [Al-Shammery, A.H. and Kalla, S.L., 2000, An extension of some hypergeometric functions of two variables. Revista de la Acade. Canaria de cie., 12, 189–196; Ben Nakhi, Y. and Kalla, S.L., 2002, A generalised beta function and associated probability density. International Journal of Mathematics and Mathematical Science, 8, 467–478.]. A study of this function will provide deeper, general and useful results in the theory of special functions and statistical distributions. A probability density function (p.d.f.) associated with this function is also introduced. The moment generating function, hazard rate function and mean residue life function for this p.d.f. will also be 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.

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.

Asymptotic expansions of the Lauricella hypergeometric function FD

The Lauricella hypergeometric function F D r ( a, b 1,…, b r ; c; x 1,…, x r ) with r∈ N , is considered for large values of one variable: x 1, or two variables: x 1 and x 2. An integral representation of this function is obtained in the form of a generalized Stieltjes transform. Distributional approach is applied to this integral to derive four asymptotic expansions of this function in increasing powers of the asymptotic variable(s) 1− x 1 or 1− x 1 and 1− x 2. For certain values of the parameters a, b i and c, two of these expansions also involve logarithmic terms in the asymptotic variable(s). For large x 1, coefficients of these expansions are given in terms of the Lauricella hypergeometric function F D r−1 ( a, b 2,…, b r ; c; x 2,…, x r ) and its derivative with respect to the parameter a, whereas for large x 1 and x 2 those coefficients are given in terms of F D r−2 ( a, b 3,…, b r ; c; x 3,…, x r ) and its derivative. All the expansions are accompanied by error bounds for the remainder at any order of the approximation. Numerical experiments show that these bounds are considerably accurate.

The geometry of the classical solutions of the Garnier systems

Our aim is to find a general approach to the theory of classical solutions of the Garnier system in n-variables, G_n, based on the Riemann-Hilbert problem and on the geometry of the space of isomonodromy deformations. Our approach consists in determining the monodromy data of the corresponding Fuchsian system that guarantee to have a classical solution of the Garnier system G_n. This leads to the idea of the reductions of the Garnier systems. We prove that if a solution of the Garnier system G_n is such that the associated Fuchsian system has l monodromy matrices equal to ±1, then it can be reduced classically to a solution of a the Garnier system with n – 1 variables G_{n – 1}. When n monodromy matrices are equal to ±1, we have classical solutions of G_n. We give also another mechanism to produce classical solutions: we show that the solutions of the Garnier systems having reducible monodromy groups can be reduced to the classical solutions found by Okamoto and Kimura in terms of Lauricella hypergeometric functions. In the case of the Garnier system in 1-variables, i.e. for the Painleve VI equation, we prove that all classical non-algebraic solutions have either reducible monodromy groups or at least one monodromy matrix equal to ±1.

Hypergeometric functions and moduli spaces of null-correlation bundles

The moduli space Nn of null-correlation bundles over CP2n+1 has a natural L2-Kähler metric. We explicitly describe that metric in terms of Appell–Lauricella hypergeometric functions. The proof uses a Penrose twistor correspondence.

Summation formulas associated with the Lauricella function FA( r)

The authors make use of some rather elementary techniques in order to derive several summation formulas associated with Lauricella's hypergeometric function F A ( r) in r variables (and also with its familiar generalizations). A number of (known or new) consequences of these summation formulas are also considered.

On the convergence of the even part of the branched continued fraction expansion of a ratio of Lauricella hypergeometric functionsF D (N)

We study the uniform and absolute convergence of the even part of the branched continued fraction that represents a ratio of Lauricella hypergeometric functions.

Approximation of the lauricella hypergeometric functionsF D (N) by branched continued fractions

Applying recursion relations for the Lauricella hypergeometric functions F NlD , we construct an expansion of a ratio of these functions in branched continued fractions. We study the convergence of the resulting expansion in the case of real parameters.

The behaviour of the fourth type of Lauricella's hypergeometric series in n variables near the boundaries of its convergence region

AbstractFor Lauricella's hypergeometric function F(n)D of n variables, we prove two formulas exhibiting its behaviour near the boundaries of the n-dimensional region of convergence of the multiple series defining it. Each of these results can be applied to deduce the corresponding properties of several simpler hypergeometric functions of one, two, and more variables.

A quantum algebra approach to basic multivariable special functions

Generalizations to a base q of the Appell and Lauricella hypergeometric functions are studied within the framework of the representation theory of quantum algebras of type An. This allows deriving various identities and formulae involving these q-functions.

Integral representations of generalized Lauricella hypergeometric functions

The generalized hypergeometric function was introduced by Srivastava and Daoust. In the present paper a new integral representation is derived. Similarly new integral representations of Lauricella and Appell function are obtained.

NONNULL DISTRIBUTION OF LRC FOR TESTING HOMOGENEITY OF COVARIANCE MATRICES OF COMPLETELY SYMMETRIC GAUSSIAN MODELS

&#x0D; &#x0D; &#x0D; The nonnull moments of the likelihood ratio statistic for testing equality of covariance matrices of completely symmetric Gaussian models are obta.ined in terms of the Lauricella's hypergeometric functions and also in terms of zonal polynomials. Then the nonnull asymptotic distribution of the statistic is derived under certain alternatives for unequal samples. &#x0D; &#x0D; &#x0D;

Neumann expansions for a certain class of generalised multiple hypergeometric series arising in physical and quantum chemical applications

A multivariable hypergeometric function which was studied recently by Niukkanen (1984) and Srivastava (1985), provides an interesting and useful unification of the generalised hypergeometric pFq function of one variable (with p numerator and q denominator parameters), Appell and Kampe de Feriet's hypergeometric functions of two variables, and Lauricella's hypergeometric functions of n variables, and also of many other classes of hypergeometric series which arise naturally in various physical and quantum chemical applications. Indeed, as already observed by Srivastava, this multivariable hypergeometric function is an obvious special case of the generalised Lauricella hypergeometric function of n variables, which was first introduced and studied systematically by Srivastava and Daoust (1969). By employing such useful connections of this 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 multivariable hypergeometric 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 number of new Neumann expansions in series of Bessel functions for the multivariable hypergeometric function from substantially more general expansions involving, for example, multiple series with essentially arbitrary terms. Some interesting special cases of the Neumann expansions presented here are also indicated.

Nonnull distribution of the likelihood ratio criterion for testing equality of covariance matrices under intraclass correlation model

The nonnull moments of the likelihood ratio statistic for testing equality of covariance matrices under intraclass correlation structure are obtained in terms of the Lauricella's hypergeometric functions and also in terms of zonal polynomials. Then the nonnull asymptotic distribution of the statistic is derived for certain alternatives.

The integration of certain products of the multivariableH-function with a general class of polynomials

The authors present six general integral formulas (four definite integrals and two contour inegrals) for theH-function of several complex variables, which was introduced and studied in a series of earlier papers by H. M. Srivastava and R. Panda (cf., e.g., [25] through [29]; see also [14] through [18], [20], [24], [32], [34], [35], [37], and [38]). Each of these integral formulas involves a product of the multivariableH-function and a general class of polynomials with essentially arbitrary coefficients which were considered elsewhere by H. M. Srivastava [21]. By assigning suiatble special values to these coefficients, the main results (contained in Theorems 1, 2 and 3 below) can be reduced to integrals involving the classical orthogonal polynomials including, for example, Hermite, Jacobi [and, of course, Gegenbauer (or ultraspherical), Legendre, and Tchebycheff], and Laguerre polynomials, the Bessel polynomials considered by H. L. Krall and O. Frink [9], and such other classes of generalized hypergeometric polynomials as those studied earlier by F. Brafman [3] and by H. W. Gould and A. T. Hopper [8]. On the other hand, the multivariableH-functions occurring in each of our main results can be reduced, under various special cases, to such simpler functions as the generalized Lauricella hypergeometric functions of several complex variables [due to H. M. Srivastava and M. C. Daoust (cf. [22] and [23])] which indeed include a great many of the useful functions (or the products of several such functions) of hypergeometric type (in one and more variables) as their particular cases (see,e. g., [1], [10] and [39]). Many of the aforementioned applications of our integral formulas (contained in Theorems 1, 2 and 3 below) are considered briefly. Further usefulness of some of these consequences of Theorems 1 and 2 in terms of the classical orthogonal polynomials is illustrated by considering a simple problem involving the orthogonal expansion of the multivariableH-function in series of Jacobi polynomials. It is also shown how these general integrals are related to a number of results scattered in the literature. 0261 0262 V

On Laplace's linear differential equation of general order

This paper discusses certain contour integral solutions of the Laplace linear differential equation of order n. It is shown, to quote one of the observations made here, how these solutions can be expressed in terms of confluent forms of Lauricella's hypergeometric function F D ( n−1) of n−1 variables.

Lauricella's hypergeometric function FD

Lauricella's hypergeometric function FD