Extended Matrix Variate Hypergeometric Functions and Matrix Variate Distributions

Daya K Nagar,Raúl Alejandro Morán-Vásquez,Arjun K Gupta

doi:10.1155/2015/190723

Abstract

Hypergeometric functions of matrix arguments occur frequently in multivariate statistical analysis. In this paper, we define and study extended forms of Gauss and confluent hypergeometric functions of matrix arguments and show that they occur naturally in statistical distribution theory.

Highlights

The classical beta function, denoted by B(a, b), is defined (Luke [1]) by the integralB (a, b) = ∫ ta−1 (1 − t)b−1 dt, Re (a) > 0, Re (b) > 0. (1)Based on the beta function, the Gauss hypergeometric function, denoted by F(a, b; c; z), and the confluent hypergeometric function, denoted by Φ(b; c; z), for Re(c) > Re(b) > 0, are defined as (Luke [1]) F (a, b; c; z) = ∫tb−1 (1 − t)c−b−1 (1 − zt)a dt, (2)󵄨󵄨󵄨󵄨arg (1 − z)󵄨󵄨󵄨󵄨 < π, Φ (b; c; z) − b)
The extended Gauss hypergeometric function and extended confluent hypergeometric function have not been generalized to the matrix case and the main objective of this work is to define these generalizations, give various integral representations, study their properties, and establish their relationships with other special functions of matrix argument
If we consider X = Im in (40) and compare the resulting expression with representation (34), we find that the extended beta function of matrix argument and extended Gauss hypergeometric function of matrix argument (EGHFMA) are connected by the expression

Summary

Introduction

The classical beta function, denoted by B(a, b), is defined (Luke [1]) by the integral. Chaudhry et al [5] found that the extended hypergeometric functions are related to the extended beta, Bessel, and Whittaker functions and gave several alternative integral representations The classical functions, such as gamma, beta, confluent hypergeometric, Gauss hypergeometric, Bessel, and Whittaker, have been generalized to the matrix case and their properties have been studied extensively. The extended Gauss hypergeometric function and extended confluent hypergeometric function have not been generalized to the matrix case and the main objective of this work is to define these generalizations, give various integral representations, study their properties, and establish their relationships with other special functions of matrix argument.

Some Known Definitions and Results

Extended Gauss Hypergeometric Function of Matrix Argument

Extended Confluent Hypergeometric Function of Matrix Argument

Extended Whittaker Function of Matrix Argument

Extended Matrix Variate Gauss Hypergeometric Function Distribution

Extended Matrix Variate Confluent Hypergeometric Function Distribution