### 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) |
---|---|

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 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

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*Linear Algebra and Its Applications*, vol. 137-138, no. C, pp. 467-478. https://doi.org/10.1016/0024-3795(90)90139-4

**Totally positive kernels, pólya frequency functions, and generalized hypergeometric series.** / Richards, Donald St P.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Totally positive kernels, pólya frequency functions, and generalized hypergeometric series

AU - Richards, Donald St P.

PY - 1990/1/1

Y1 - 1990/1/1

N2 - 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) = pFq(xy), x,y ε{lunate} R, wherepFq 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 ai > 0 and ki is a positive integer (i = 1,...,p) then K(x,y) = pFq(a1,...,ap;a1 + k1,...,ap + kp; xy) is strictly totally positive on R2. In the second part, we use the theory of entire functions to derive some Pólya frequency function properties of the hypergeometric series pFq(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 ai and ki are as above, then there exists a probability density function f on R, such that f is a strict Pólya frequency function, and 1/pFp(a1+k1,...,ap +kp;a1,...,ap;z=Lf(z), the Laplace transform of f.

AB - 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) = pFq(xy), x,y ε{lunate} R, wherepFq 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 ai > 0 and ki is a positive integer (i = 1,...,p) then K(x,y) = pFq(a1,...,ap;a1 + k1,...,ap + kp; xy) is strictly totally positive on R2. In the second part, we use the theory of entire functions to derive some Pólya frequency function properties of the hypergeometric series pFq(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 ai and ki are as above, then there exists a probability density function f on R, such that f is a strict Pólya frequency function, and 1/pFp(a1+k1,...,ap +kp;a1,...,ap;z=Lf(z), the Laplace transform of f.

UR - http://www.scopus.com/inward/record.url?scp=34247369226&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34247369226&partnerID=8YFLogxK

U2 - 10.1016/0024-3795(90)90139-4

DO - 10.1016/0024-3795(90)90139-4

M3 - Article

AN - SCOPUS:34247369226

VL - 137-138

SP - 467

EP - 478

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - C

ER -