### Abstract

In multivariate statistical analysis, several authors have studied the total positivity properties of the generalized ( _{0}F _{1}) hypergeometric function of two real symmetric matrix arguments. In this paper, we make use of zonal polynomial expansions to obtain a new proof of a result that these _{0}F _{1} functions fail to satisfy certain pairwise total positivity properties; this proof extends both to arbitrary generalized ( _{r}F _{s}) functions of two matrix arguments and to the generalized hypergeometric functions of Hermitian matrix arguments. In the case of the generalized hypergeometric functions of two Hermitian matrix arguments, we prove that these functions satisfy certain modified pairwise TP _{2} properties; the proofs of these results are based on Sylvester's formula for compound determinants and the condensation formula of C. L. Dodgson [Lewis Carroll] (1866).

Original language | English (US) |
---|---|

Pages (from-to) | 907-922 |

Number of pages | 16 |

Journal | Journal of Statistical Physics |

Volume | 116 |

Issue number | 1-4 |

DOIs | |

State | Published - Aug 1 2004 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

}

*Journal of Statistical Physics*, vol. 116, no. 1-4, pp. 907-922. https://doi.org/10.1023/B:JOSS.0000037249.50382.92

**Total positivity properties of generalized hypergeometric functions of matrix argument.** / Richards, Donald.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Total positivity properties of generalized hypergeometric functions of matrix argument

AU - Richards, Donald

PY - 2004/8/1

Y1 - 2004/8/1

N2 - In multivariate statistical analysis, several authors have studied the total positivity properties of the generalized ( 0F 1) hypergeometric function of two real symmetric matrix arguments. In this paper, we make use of zonal polynomial expansions to obtain a new proof of a result that these 0F 1 functions fail to satisfy certain pairwise total positivity properties; this proof extends both to arbitrary generalized ( rF s) functions of two matrix arguments and to the generalized hypergeometric functions of Hermitian matrix arguments. In the case of the generalized hypergeometric functions of two Hermitian matrix arguments, we prove that these functions satisfy certain modified pairwise TP 2 properties; the proofs of these results are based on Sylvester's formula for compound determinants and the condensation formula of C. L. Dodgson [Lewis Carroll] (1866).

AB - In multivariate statistical analysis, several authors have studied the total positivity properties of the generalized ( 0F 1) hypergeometric function of two real symmetric matrix arguments. In this paper, we make use of zonal polynomial expansions to obtain a new proof of a result that these 0F 1 functions fail to satisfy certain pairwise total positivity properties; this proof extends both to arbitrary generalized ( rF s) functions of two matrix arguments and to the generalized hypergeometric functions of Hermitian matrix arguments. In the case of the generalized hypergeometric functions of two Hermitian matrix arguments, we prove that these functions satisfy certain modified pairwise TP 2 properties; the proofs of these results are based on Sylvester's formula for compound determinants and the condensation formula of C. L. Dodgson [Lewis Carroll] (1866).

UR - http://www.scopus.com/inward/record.url?scp=4344598188&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4344598188&partnerID=8YFLogxK

U2 - 10.1023/B:JOSS.0000037249.50382.92

DO - 10.1023/B:JOSS.0000037249.50382.92

M3 - Article

VL - 116

SP - 907

EP - 922

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1-4

ER -