Donald St. P. Richards

Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions

Hypergeometric functions of matrix argument arise in a diverse range of applications in harmonic analysis, multivariate statistics, quantum physics, molecular chemistry, and number theory. This paper presents a general theory of such functions for real division algebras. These functions, which generalize the classical hypergeometric functions, are defined by infinite series on the space S = S(n, F) of all n x n Hermitian matrices over the division algebra F. The theory depends intrinsically upon the representation theory of the general linear group G = GL(n, F) of invertible n x n matrices over F, and the theme of this work is the full exploitation of the inherent group theory. The main technique is the use of the method of algebraic induction to realize explicitly the appropriate representations of G, to decompose the space of polynomial functions on S, and to describe the zonal polynomials from which the hypergeometric functions are constructed. Detailed descriptions of the convergence properties of the series expansions are given, and integral representations are provided. Future papers in this series will develop the fine structure of these functions. 0. Introduction. We begin a series of articles in which we develop the fine structure of generalized hypergeometric functions of matrix argument. By fine structure, we allude to the analogues of such classical results as series expansions, integral formulas, asymptotics, differential equations, summation formulas, addition theorems, composition formulas, and recurrence relations. This first paper, in which we simultaneously treat real, complex, and quaternionic analysis, is the result of our desire to present a complete theory of hypergeometric functions of matrix argument over real division algebras, not only as a framework for the body of detailed results to follow in later papers, but also to clarify the representationtheoretic foundation for such a theory. Although these hypergeometric functions are of interest on purely analytic grounds, they arise in a wide range of applications. Indeed, various classes of Received by the editors May 19, 1986. 1980 Malhema1zics S*ject Claszficain (1985 Ren). Primary 22E30, 22E45, 33A75, 43A85, 43A90, 62H10; Secondary 20G20, 32A07, 32M15, 44A10, 62E15. Key w and phrases. Generalized hypergeometric functions, zonal polynomials, representation theory, algebraic induction, multivariate statistics, general linear group, generalized gamma functions, Pochhammer symbols, Laplace transforms, maximal compact subgroup, invariant polynomials, positive cones, symmetric spaces, Schur functions, special functions of matrix argument. The first author is on leave from the University of Wyoming. The second author was partially supported by the National Science Foundation under Grant MCS-8403381, and by the Research Council of the University of North Carolina. (B1987 American Mathematical Society 0002-9947/87

Multivariate Liouville distributions, III

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Positive definite symmetric functions on finite-dimensional spaces II

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Approximating the matrix Fisher and Bingham distributions: applications to spherical regression and Procrustes analysis

A random vector (X1, ..., Xn), with positive components, has a Liouville distribution if its joint probability density function is of the formf(x1 + ... + xn)x1^a^^1^.^1 ... xn^a^^n^.^1 with theai all positive. Examples of these are the Dirichlet and inverted Dirichlet distributions. In this paper, a comprehensive treatment of the Liouville distributions is provided. The results pertain to stochastic representations, transformation properties, complete neutrality, marginal and conditional distributions, regression functions, and total positivity and reverse rule properties. Further, these topics are utilized in various characterizations of the Dirichlet and inverted Dirichlet distributions. Matrix analogs of the Liouville distributions are also treated, and many of the results obtained in the vector setting are extended appropriately.

Positive definite symmetric functions on finite dimensional spaces. I. Applications of the Radon transform

Integral representations for the density functions of absolutely continuous [alpha]-symmetric random vectors are derived, and general methods for constructing new [alpha]-symmetric distributions are presented. An explicit formula, for determining the spectral measure of a symmetric stable random vector from its characteristic function, is obtained.

Hypergeometric functions on complex matrix space

We obtain approximations to the distribution of the exponent in the matrix Fisher distributions on SO(p) and on O(p) whose density with respect to Haar measure is proportional to exp(Tr GX0tX). Similar approximations are found for the distribution of the exponent in the Bingham distribution, with density proportional to exp(xtGx), on the unit sphere Sp-1 in Euclidean p-dimensional space. The matrix Fisher distribution arises as the exact conditional distribution of the maximum likelihood estmate of the unknown orthogonal matrix in the spherical regression model on Sp-1 with Fisher distributed errors. It also arises as the exact conditional distribution of the maximum likelihood estimate of the unknown orthogonal matrix in a model of Procrustes analysis in which location and orientation, but not scale, changes are allowed. These methods allow determination of a confidence region for the unknown rotation for moderate sample sizes with moderate error concentrations when the error concentration parameter is known.

Positivity of integrals of Bessel functions

An n-dimensional random vector X is said (Cambanis, S., Keener, R., and Simons, G. (1983). J. Multivar. Anal., 13 213-233) to have an [alpha]-symmetric distribution, [alpha] > 0, if its characteristic function is of the form [phi]([xi]1[alpha] + ... + [xi]n[alpha]). Using the Radon transform, integral representations are obtained for the density functions of certain absolutely continuous [alpha]-symmetric distributions. Series expansions are obtained for a class of apparently new special functions which are encountered during this study. The Radon transform is also applied to obtain the densities of certain radially symmetric stable distributions on n. A new class of zonally symmetric stable laws on n is defined, and series expansions are derived for their characteristic functions and densities.

The central limit theorem on spaces of positive definite matrices

The authors had presented the general foundations for a theory of hypergeometric functions of matrix argument over real division algebras. Here, they further develop the fine structure of these functions over the complex field, including series expansions, integral representations, asymptotics, differential equations, addition formulas, multiplication formulas, summation theorems, transformation properties, etc. Especially important are the operator-valued hypergeometric functions, required for (nonspherical) expansions such as addition formulas by the noncommutativity of matrix multiplication

Hypergeometric Functions of Scalar Matrix Argument are Expressible in Terms of Classical Hypergeometric Functions

Using results on multiply monotone functions, we establish the positivity of integrals of Bessel functions of the form [ int_0^x {left( {x^mu - t^mu } right)^lambda } t^alpha J_beta (t)dt,quad x > 0,] where

Totally positive kernels, pólya frequency functions, and generalized hypergeometric series

0 < mu leq 1 leq lambda