The main aim of this paper is to give the integral representations for the so called multidimensional negative definite functions defined on the product of commutative hypergroups. Some necessary and sufficient conditions that guarantee, the product of multidimensional positive definite functions defined on the product of hypergroups is also multidimensional positive definite, are obtained. Moreover, many relations that governed the relations between the classes of multidimensional positive and negative definite functions are shown. Also, we prove that a continuous function with compact support ? is multidimensional negative definite on the product of hypergroups, if and only if exp(?t?) is multidimensional positive definite for each t & 0.
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