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

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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

All Science Journal Classification (ASJC) codes

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

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