QUANTUM HOMOGENEOUS SPACES AND SPECIAL-FUNCTIONS WITH A DIMENSIONAL DEFORMATION PARAMETER

We study the most elementary aspects of harmonic analysis on a homogen eous space of a deformation of the two-dimensional Euclidean group, ad mitting generalizations to dimensions three and four, whose quantum pa rameter has the physical dimensions of length. The homogeneous space i s recognized as a new quantum plane and the action of the Euclidean qu antum group is used to determine an eigenvalue problem for the Casimir operator, which constitutes the analogue of the Schrodinger equation in the presence of such a deformation. The solutions are given in the plane-wave and angular-momentum bases and are expressed in terms of hy pergeometric series with non-commuting parameters.