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Total positivity properties of generalized hypergeometric functions of matrix argument. (English) Zbl 1134.33315
Summary: In multivariate statistical analysis, several authors have studied the total positivity properties of the generalized ${\left(}_{0}{F}_{1}\right)$ 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 ${\left(}_{r}{F}_{s}\right)$ 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 $T{P}_{2}$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).
##### MSC:
 33C70 Other hypergeometric functions and integrals in several variables 33C20 Generalized hypergeometric series, ${}_{p}{F}_{q}$ 33E20 Functions defined by series and integrals 43A90 Spherical functions (abstract harmonic analysis) 62H99 Multivariate analysis 60E15 Inequalities in probability theory; stochastic orderings 15A52 Random matrices (MSC2000)