Certain hypergeometric series related to the root system 𝐵𝐶

Beerends, R. J.; Opdam, E. M.

doi:10.1090/S0002-9947-1993-1123450-8

Certain hypergeometric series related to the root system $BC$
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by R. J. Beerends and E. M. Opdam PDF

Trans. Amer. Math. Soc. 339 (1993), 581-609 Request permission

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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