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We consider the hypergeometric translation operators associated to the Cherednik operators and the Heckman–Opdam theory onℝ  d${\mathbb{R}^{d}}$    introduced by the author in [ 6 ]. Under some conditions on the root system and the multiplicity function, we prove in this paper that these operators are positivity preserving and allow positive integral representations. In particular, we deduce for the Opdam–Cherednik kernel and the Heckman–Opdam hypergeometric function the following main results:

The positivity of the hypergeometric translation operators associated to the Cherednik operators and the Heckman--Opdam theory on ℝ d