Matrix hypergeometric function and its application to computation of characteristic function of spherically symmetric distributions with phase-type amplitude

Authors:

• Marek Nałęcz

Abstract

The paper presents a new definition of a matrix-valued Gauss hypergeometric function2F1of a matrix argument. Theproposed function is applied in the context of computing a characteristic function for spherically symmetric randomvectors whose amplitude has phase-type distribution. It is proved that the characteristic function can be expressedas a specialized case of the Gauss hypergeometric function with the argument related to a matrix representation ofthe phase-type distribution. Then it is shown that this specialized case of the2F1function can be evaluated by usingclosed-form formulas, which involve only elementary functions and thus can reliably be computed numerically fora matrix argument. Explicit formulas are derived for the moments of the spherically symmetric distribution underconsideration. As a by-product of the research, an erratum to the tables of special functions is proposed.