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Some time ago, Rideau and Winternitz introduced a realization of the quantumalgebra su_q(2) on a real two-dimensional sphere, or a real plane, andconstructed a basis for its representations in terms of q-special functions,which can be expressed in terms of q-Vilenkin functions, and are related tolittle q-Jacobi functions, q-spherical functions, and q-Legendre polynomials.In their study, the values of q were implicitly restricted to $q \in R^+$. Inthe present paper, we extend their work to the case of generic values of $q \inS^1$ (i.e., q values different from a root of unity). In addition, we unitarizethe representations for both types of q values, $q \in R^+$ and generic $q \inS^1$, by determining some appropriate scalar products. From the latter, wededuce the orthonormality relations satisfied by the q-Vilenkin functions.Comment: 24 pages, LaTeX, uses amssym, 1 PostScript figur

Unitary representations of the quantum algebra suq(2) on a real two-dimensional sphere for q∈R+ or generic q∈S1