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

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 ${}_1{F}_0^{\beta }(a;\mathbf{Z})={}_1^k{P}_0^{\beta ,1}(1:a;\mathbf{Z})=\mathrm{\mid }\mathbf{I}-\mathbf{Z}{\mathrm{\mid }}^{-a}$ 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.