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

*Marek Nałęcz*

*Statistics & Probability Letters*, 2021, vol. 173, issue C

**Abstract:**
The paper presents a new definition of a matrix-valued Gauss hypergeometric function 2F1 of a matrix argument. The proposed function is applied in the context of computing a characteristic function for spherically symmetric random vectors whose amplitude has phase-type distribution. It is proved that the characteristic function can be expressed as a specialized case of the Gauss hypergeometric function with the argument related to a matrix representation of the phase-type distribution. Then it is shown that this specialized case of the 2F1 function can be evaluated by using closed-form formulas, which involve only elementary functions and thus can reliably be computed numerically for a matrix argument. Explicit formulas are derived for the moments of the spherically symmetric distribution under consideration. As a by-product of the research, an erratum to the tables of special functions is proposed.

**Keywords:** Spherically symmetric random vectors; Phase-type distributions; Characteristic function; Special functions; Generalized Gauss hypergeometric function (search for similar items in EconPapers)

**Date:** 2021

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**Persistent link:** https://EconPapers.repec.org/RePEc:eee:stapro:v:173:y:2021:i:c:s0167715221000249

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**DOI:** 10.1016/j.spl.2021.109062

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