### Abstract

Recently, K. I. Gross and the author [J. Approx. Theory 59:224-246 (1989)] analyzed the total positivity of kernels of the form K(x,y) = _{p}F_{q}(xy), x,y ε{lunate} R, where_{p}F_{q} denotes a classical generalized hypergeometric series. There, we related the determinants which define the total positivity of K to the hypergeometric functions of matrix argument, defined on the space of n × n Hermitian matrices. In the first part of this paper, we apply these methods to derive the total positivity properties of the kernels K for more general choices of the parameters in these hypergeometric series. In particular, we prove that if a_{i} > 0 and k_{i} is a positive integer (i = 1,...,p) then K(x,y) = _{p}F_{q}(a_{1},...,a_{p};a_{1} + k_{1},...,a_{p} + k_{p}; xy) is strictly totally positive on R^{2}. In the second part, we use the theory of entire functions to derive some Pólya frequency function properties of the hypergeometric series _{p}F_{q}(x). Some of these latter results suggest a curious duality with the total positivity properties described above. For instance, we prove that if p > 1 and the a_{i} and k_{i} are as above, then there exists a probability density function f on R, such that f is a strict Pólya frequency function, and 1/_{p}F_{p}(a_{1}+k_{1},...,a_{p }+k_{p};a_{1},...,a_{p};z=Lf(z), the Laplace transform of f.

Original language | English (US) |
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Pages (from-to) | 467-478 |

Number of pages | 12 |

Journal | Linear Algebra and Its Applications |

Volume | 137-138 |

Issue number | C |

DOIs | |

State | Published - Jan 1 1990 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics