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 ${}_{1}F_{0}^{\beta}(a;\mathbf{Z})={}_{1}^{k}P_{0}^{\beta,1}(1:a;\mathbf{Z})=|\mathbf{I}-\mathbf{Z}|^{-a}$ 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.