Abstract
This paper is devoted to give the integral representations for some functions related to multidimensional positive definite functions defined on the product of commutative hypergroups. Also, we prove that a continuous function with compact support psi is multidimensional conditionally exponential convex on the product of commutative hypergroups if and only if exp(-t psi) is multidimensional conditionally exponential convex for each t> 0. Moreover, we give some relations between the class of multidimensional completely monotonic functions and the set of multidimensional tau-positive functions on the product of commutative hypergroups. Finally, we find the integral representations of multidimensional completely monotonic functions and multidimensional tau-positive functions on the product of commutative hypergroups. (C) 2016 Physics Essays Publication.