Hypergeometric Functions I

@article{MacDonald2013HypergeometricFI,
title={Hypergeometric Functions I},
author={Ian G. MacDonald},
journal={arXiv: Classical Analysis and ODEs},
year={2013}
}
• I. MacDonald
• Published 18 September 2013
• Mathematics
• arXiv: Classical Analysis and ODEs
This is the typewritten version of a handwritten manuscript which was completed by Ian G. Macdonald in 1987 or 1988. The manuscript is a very informal working paper, never intended for formal publication. Nevertheless, copies of the manuscript have circulated widely, giving rise to quite a few citations in the subsequent 25 years. Therefore it seems justified to make the manuscript available for the whole mathematical community. The author kindly gave his permission that a typewritten version…
Riesz distributions and Laplace transform in the Dunkl setting of type A
We study Riesz distributions in the framework of rational Dunkl theory associated with root systems of type A. As an important tool, we employ a Laplace transform involving the associated Dunkl
Branching Rules for Symmetric Hypergeometric Polynomials
• Mathematics, Physics
• 2016
Starting from a recently found branching formula for the six-parameter family of symmetric Macdonald-Koornwinder polynomials, we arrive by degeneration at corresponding branching rules for symmetric
C 2013 Society for Industrial and Applied Mathematics Eigenvalue Distributions of Beta-wishart Matrices *
We derive explicit expressions for the distributions of the extreme eigenvalues of the Beta-Wishart random matrices in terms of the hypergeometric function of a matrix argument. These results
Okounkov's BC-type interpolation Macdonald polynomials and their q=1 limit
This paper surveys eight classes of polynomials associated with $A$-type and $BC$-type root systems: Jack, Jacobi, Macdonald and Koornwinder polynomials and interpolation (or shifted) Jack and
Eigenvalue distributions of beta-Wishart matrices
• Mathematics
• 2014
We derive explicit expressions for the distributions of the extreme eigenvalues of the beta-Wishart random matrices in terms of the hypergeometric function of a matrix argument. These results
Eigenvalue statistics for the sum of two complex Wishart matrices
The sum of independent Wishart matrices, taken from distributions with unequal covariance matrices, plays a crucial role in multivariate statistics, and has applications in the fields of quantitative
Universal Behavior of the Corners of Orbital Beta Processes
• Cesar Cuenca
• Mathematics, Physics
International Mathematics Research Notices
• 2019
There is a unique unitarily-invariant ensemble of $N\times N$ Hermitian matrices with a fixed set of real eigenvalues $a_1> \dots > a_N$. The joint eigenvalue distribution of the $(N-1)$ top-left
Beta Distributions and Sonine Integrals for Bessel Functions on Symmetric Cones
• Mathematics
• 2018
There exist several multivariate extensions of the classical Sonine integral representation for Bessel functions of some index $\mu+ \nu$ with respect to such functions of lower index $\mu.$ For
Positive intertwiners for Bessel functions of type B
• Mathematics
• 2019
Let $V_k$ denote Dunkl's intertwining operator for the root sytem $B_n$ with multiplicity $k=(k_1,k_2)$ with $k_1\geq 0, k_2>0$. It was recently shown that the positivity of the operator
PROPERTIES OF THE INVERSE OF A NONCENTRAL WISHART MATRIX
• Mathematics
• 2021
The inverse of a noncentral Wishart matrix occurs in a variety of contexts in multivariate statistical work, including instrumental variables (IV) regression, but there has been very little work on

References

SHOWING 1-9 OF 9 REFERENCES
BESSEL FUNCTIONS OF MATRIX ARGUMENT
Our principal results fall into three main classes. First, a large number of formulae from the classical theory of special functions are given appropriate generalizations. Some of these turn out to
Partial differential equations for hypergeometric functions of two argument matrices
• Mathematics
• 1972
In multivariate analysis many of the noncentral latent root distributions can be expressed in terms of hypergeometric functions vFq of two-argument matrices. This paper is concerned with showing that
Partial differential equations for hypergeometric functions 3F2 of matrix argument
Many multivariate non-null distributions and moment formulas can be expressed in terms of hypergeometric functions pFq of matrix arqument. Muirhead [6] and Constantine and Muirhead [2] gave partial
Special Functions of Matrix and Single Argument in Statistics
Publisher Summary This chapter discusses some special functions of matrix and single argument in statistics. The zonal spherical functions are related to certain symmetric spaces and to the
Multivariate calculation
• 1985
Multivariate calculation Springer Series in Statistics
• Multivariate calculation Springer Series in Statistics
• 1985
Partial differential equations for hypergeometric functions 3 F 2 of matrix argument