Total positivity, spherical series, and hypergeometric functions of matrix argument

Kenneth I. Gross, Donald St P. Richards

Research output: Contribution to journalArticlepeer-review

104 Scopus citations


Given a totally positive function K of two real variables, is there a method for establishing the total positivity of K in an "obvious" fashion? In the case in which K(x, y) = f(xy), where f is real-analytic in a neighborhood of zero, we obtain integral representations for the determinants which define the total positivity of K. The total positivity of K then follows immediately from positivity of the integrands. In particular, we analyze the total positivity of classical hypergeometric functions by these methods. The central theme of this work is the circle of ideas that relates total positivity to "spherical series" on the symmetric space GL(n, C) U(n), and classical hypergeometric functions to hypergeometric functions of matrix argument.

Original languageEnglish (US)
Pages (from-to)224-246
Number of pages23
JournalJournal of Approximation Theory
Issue number2
StatePublished - Nov 1989

All Science Journal Classification (ASJC) codes

  • Analysis
  • Numerical Analysis
  • General Mathematics
  • Applied Mathematics


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