We define the -hypergeometric functions as a generalization of the hypergeometric functions associated with root systems of Heckman and Opdam. In the geometric setting, the -hypergeometric functions can be specialized to Harish-Chandras 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 -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.