Asymptotic analysis of Θ-hypergeometric functions

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.