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Certain hypergeometric series related to the root system $ BC$


Authors: R. J. Beerends and E. M. Opdam
Journal: Trans. Amer. Math. Soc. 339 (1993), 581-609
MSC: Primary 33C80; Secondary 05E35, 22E70
DOI: https://doi.org/10.1090/S0002-9947-1993-1123450-8
MathSciNet review: 1123450
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Abstract: We show that the generalized hypergeometric function $ _2{F_1}$ of matrix argument is the series expansion at the origin of a special case of the hypergeometric function associated with the root system of type $ BC$. In addition we prove that the Jacobi polynomials of matrix argument correspond to the Jacobi polynomials associated with the root system of type $ BC$. We also give a precise relation between Jack polynomials and the Jacobi polynomials associated with the root system of type $ A$. As a side result one obtains generalized hook-length formulas which are related to Harish-Chandra's $ {\mathbf{c}}$-function and one can prove a conjecture due to Macdonald relating two inner products on a space of symmetric functions.


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DOI: https://doi.org/10.1090/S0002-9947-1993-1123450-8
Article copyright: © Copyright 1993 American Mathematical Society

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