### Abstract

We consider an N-dimensional real integral, indexed by a parameter that specifies the power of a Vandermonde determinant. For two particular values of the parameter, this integral arises from matrix integrals, over real symmetric and complex Hermitian N × N matrices. When it is normalized, it gives the expectation of an arbitrary power of the determinant. The results are given as finite summations, using terminating hypergeometric series. We relate the integral to a specific coefficient in the Jack polynomial indexed by a partition of rectangular shape, and present data for this coefficient in terms of the parameter α.

Original language | English (US) |
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Pages (from-to) | 377-390 |

Number of pages | 14 |

Journal | Studies in Applied Mathematics |

Volume | 110 |

Issue number | 4 |

DOIs | |

State | Published - May 1 2003 |

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### All Science Journal Classification (ASJC) codes

- Applied Mathematics

### Cite this

*Studies in Applied Mathematics*,

*110*(4), 377-390. https://doi.org/10.1111/1467-9590.00243

}

*Studies in Applied Mathematics*, vol. 110, no. 4, pp. 377-390. https://doi.org/10.1111/1467-9590.00243

**Determinants of random matrices and jack polynomials of rectangular shape.** / Andrews, G. E.; Goulden, I. P.; Jackson, D. M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Determinants of random matrices and jack polynomials of rectangular shape

AU - Andrews, G. E.

AU - Goulden, I. P.

AU - Jackson, D. M.

PY - 2003/5/1

Y1 - 2003/5/1

N2 - We consider an N-dimensional real integral, indexed by a parameter that specifies the power of a Vandermonde determinant. For two particular values of the parameter, this integral arises from matrix integrals, over real symmetric and complex Hermitian N × N matrices. When it is normalized, it gives the expectation of an arbitrary power of the determinant. The results are given as finite summations, using terminating hypergeometric series. We relate the integral to a specific coefficient in the Jack polynomial indexed by a partition of rectangular shape, and present data for this coefficient in terms of the parameter α.

AB - We consider an N-dimensional real integral, indexed by a parameter that specifies the power of a Vandermonde determinant. For two particular values of the parameter, this integral arises from matrix integrals, over real symmetric and complex Hermitian N × N matrices. When it is normalized, it gives the expectation of an arbitrary power of the determinant. The results are given as finite summations, using terminating hypergeometric series. We relate the integral to a specific coefficient in the Jack polynomial indexed by a partition of rectangular shape, and present data for this coefficient in terms of the parameter α.

UR - http://www.scopus.com/inward/record.url?scp=0037725328&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037725328&partnerID=8YFLogxK

U2 - 10.1111/1467-9590.00243

DO - 10.1111/1467-9590.00243

M3 - Article

AN - SCOPUS:0037725328

VL - 110

SP - 377

EP - 390

JO - Studies in Applied Mathematics

JF - Studies in Applied Mathematics

SN - 0022-2526

IS - 4

ER -