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Some time ago, Rideau and Winternitz introduced a realization of the quantum algebra suq(2) on a real two-dimensional sphere, or a real plane, and constructed a basis for its representations in terms of q-special functions, which can be expressed in terms of q-Vilenkin functions. In their study, the values of q were implicitly restricted to q ∈ R+. In the present paper, we extend their work to the case of generic values of q ∈ S1 (i.e., q values different from a root of unity). In addition, we unitarize the representations for both types of q values, q ∈ R+ and generic q ∈ S1, by determining some appropriate scalar products.