Multidimensional positive definite functions on the product of hypergroups

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 ψ is multidimensional conditionally exponential convex on the product of commutative hypergroups if and only if exp(−tψ) 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 τ-positive functions on the product of commutative hypergroups. Finally, we find the integral representations of multidimensional completely monotonic functions and multidimensional τ-positive functions on the product of commutative hypergroups.