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Asymptotic analysis of Θ-hypergeometric functions

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Abstract

We define the Θ-hypergeometric functions as a generalization of the hypergeometric functions associated with root systems of Heckman and Opdam. In the geometric setting, the Θ-hypergeometric functions can be specialized to Harish-Chandra’s spherical functions on Riemannian symmetric spaces of noncompact type, and also to the spherical functions on noncompactly causal symmetric spaces. After describing their regularity properties, we prove estimates for the Θ-hypergeometric functions which are uniform in the space parameter and locally uniform in the spectral parameter. Particular cases are sharp uniform estimates for the Harish-Chandra series up to the walls of the positive Weyl chamber. New estimates for the spherical functions on noncompactly causal symmetric spaces are deduced.

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Authors and Affiliations

  1. Institut für Mathematik, TU-Clausthal, 38678, Clausthal-Zellerfeld, Germany

    Angela Pasquale

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  1. Angela Pasquale
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Mathematics Subject Classification (2000)

33C67, 43A90, 43A85

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Pasquale, A. Asymptotic analysis of Θ-hypergeometric functions. Invent. math. 157, 71–122 (2004). https://doi.org/10.1007/s00222-003-0349-9

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  • DOI: https://doi.org/10.1007/s00222-003-0349-9

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