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Our main goal is to determine the continuous and bounded complex valued solutions of the functional equationwhere X is a hypergroup. The solutions are expressed in terms of 2 -dimensional representations of X. The papers of Davison [10] and Stetkaer [25,26] are the essential motivation for this first part of the present work and the methods used here are closely related to and inspired by those in [10,25,26]. In addition, superstability problem for this functional equation on any hypergroup and without any condition on f is considered.

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