### Abstract

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 language | English (US) |
---|---|

Pages (from-to) | 224-246 |

Number of pages | 23 |

Journal | Journal of Approximation Theory |

Volume | 59 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1989 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Numerical Analysis
- Mathematics(all)
- Applied Mathematics

### Cite this

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*Journal of Approximation Theory*, vol. 59, no. 2, pp. 224-246. https://doi.org/10.1016/0021-9045(89)90153-6

**Total positivity, spherical series, and hypergeometric functions of matrix argument.** / Gross, Kenneth I.; Richards, Donald.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Total positivity, spherical series, and hypergeometric functions of matrix argument

AU - Gross, Kenneth I.

AU - Richards, Donald

PY - 1989/1/1

Y1 - 1989/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0001385789&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001385789&partnerID=8YFLogxK

U2 - 10.1016/0021-9045(89)90153-6

DO - 10.1016/0021-9045(89)90153-6

M3 - Article

AN - SCOPUS:0001385789

VL - 59

SP - 224

EP - 246

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

SN - 0021-9045

IS - 2

ER -