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8 - Jacobi Polynomials and Hypergeometric Functions Associated with Root Systems
- By G. J. Heckman, E. M. Opdam
Edited by Tom H. Koornwinder, Universiteit van Amsterdam, Jasper V. Stokman, Universiteit van Amsterdam
Book:

Encyclopedia of Special Functions: The Askey-Bateman Project

Published online:

30 September 2020

Print publication:

15 October 2020, pp 217-257
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Summary

In this chapter the basic theory of hypergeometric functions for root systems is discussed, together with some of its applications. These functions arise in various contexts such as the representation theory of reductive groups, multivariate orthogonal polynomials, harmonic analysis on symmetric spaces, integrable models in mathematical physics, Dunkl operators, Knizhnik-Zamolodchikov connections, affine Hecke algebras, algebraic combinatorics (e.g. the Macdonald conjectures), the determination of certain Bernstein-Sato polynomials and period integrals for moduli of Del Pezzo surfaces.