Matrix generalized elliptical binomial series under real normed division algebras and the central matrix variate beta distribution

TL;DR

This paper extends the scalar binomial series to matrices within real normed division algebras, providing new series representations and deriving the matrix variate beta distribution under elliptical models across various algebraic systems.

Contribution

It introduces a matrix generalization of the binomial series under elliptical models and real normed division algebras, unifying and extending existing scalar and matrix distributions.

Findings

Derived a matrix binomial series under elliptical models.

Unified the matrix variate beta distribution across different algebraic systems.

Provided series representations for hypergeometric functions in matrix form.

Abstract

In this paper we provide a matrix extension of the scalar binomial series under elliptical contoured models and real normed division algebras. The classical hypergeometric series ${}_{1}F_{0}^{\beta}(a;\mathbf{Z})={}_{1}^{k}P_{0}^{\beta,1}(1:a;\mathbf{Z})=|\mathbf{I}-\mathbf{Z}|^{-a}$ of Jack polynomials are now seen as an invariant generalized determinant with a series representation indexed by any elliptical generator function. In particular, a corollary emerges for a simple derivation of the central matrix variate beta type II distribution under elliptically contoured models in the unified real, complex, quaternions and octonions.