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

Research output: Contribution to journalArticle

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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) = 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.

Original languageEnglish (US)
Pages (from-to)467-478
Number of pages12
JournalLinear Algebra and Its Applications
Volume137-138
Issue numberC
DOIs
StatePublished - Jan 1 1990

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Generalized Hypergeometric Series
Total Positivity
Probability density function
kernel
Hypergeometric Series
Laplace transforms
Hypergeometric Functions
Hermitian matrix
Entire Function
Gross
Laplace transform
Determinant
Duality
Strictly
Denote
Integer

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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title = "Totally positive kernels, p{\'o}lya frequency functions, and generalized hypergeometric series",
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) = 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 P{\'o}lya 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 P{\'o}lya frequency function, and 1/pFp(a1+k1,...,ap +kp;a1,...,ap;z=Lf(z), the Laplace transform of f.",
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Totally positive kernels, pólya frequency functions, and generalized hypergeometric series. / Richards, Donald St P.

In: Linear Algebra and Its Applications, Vol. 137-138, No. C, 01.01.1990, p. 467-478.

Research output: Contribution to journalArticle

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