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elliptic selberg integrals

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6 - Elliptic Hypergeometric Functions Associated with Root Systems
- By H. Rosengren, S. O. Warnaar
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 159-186
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Summary

We give a survey of elliptic hypergeometric functions associated with root systems, comprised of three main parts. The first two in essence form an annotated table of the main evaluation and transformation formulas for elliptic hypergeometric integrals and series on root systems. The third and final part gives an introduction to Rains' elliptic Macdonald-Koornwinder theory (in part also developed by Coskun and Gustafson). We survey the main properties of elliptic BC_n interpolation functions and BC_n-symmetric biorthogonal functions, which generalize Okounkov's BC_n interpolation Macdonald polynomials and the Koornwinder polynomials, respectively.