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Integral representation and uniform limits for some Heckman-Opdam hypergeometric functions of type BC


Authors: Margit Rösler and Michael Voit
Journal: Trans. Amer. Math. Soc. 368 (2016), 6005-6032
MSC (2010): Primary 33C67, 43A90; Secondary 33C52, 22E46
DOI: https://doi.org/10.1090/tran6673
Published electronically: June 17, 2015
MathSciNet review: 3458405
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Abstract: The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi functions in one variable and include the spherical functions of non-compact Grassmann manifolds over the real, complex or quaternionic numbers. There are various limit transitions known for such hypergeometric functions. In the present paper, we use an explicit form of the Harish-Chandra integral representation as well as an interpolated variant in order to obtain two limit results, each of them for three continuous classes of hypergeometric functions of type BC which extend the group cases over the fields $ \mathbb{R}, \mathbb{C}, \mathbb{H}.$ These limits are distinguished from the known results by explicit and uniform error bounds. The first limit realizes the approximation of the spherical functions of infinite dimensional Grassmannians of fixed rank; here hypergeometric functions of type A appear as limits. The second limit is a contraction limit towards Bessel functions of Dunkl type.


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Additional Information

Margit Rösler
Affiliation: Institut für Mathematik, Universität Paderborn, Warburger Str. 100, D-33098 Paderborn, Germany
Email: roesler@math.upb.de

Michael Voit
Affiliation: Fakultät Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, D-44221 Dortmund, Germany
Email: michael.voit@math.tu-dortmund.de

DOI: https://doi.org/10.1090/tran6673
Received by editor(s): February 27, 2014
Received by editor(s) in revised form: January 19, 2015
Published electronically: June 17, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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