  We study the most elementary aspects of harmonic analysis on a homogeneous
space of a deformation of the two-dimensional Euclidean group, admitting
generalizations to dimensions three and four, whose quantum parameter has the
physical dimensions of a length. The homogeneous space is recognized as a new
quantum plane and the action of the Euclidean quantum group is used to
determine an eigenvalue problem for the Casimir operator, that constitutes the
analogue of the Schroedinger equation in the presence of such deformation. The
solutions are given in the plane wave and in the angular momentum bases and are
expressed in terms of hypergeometric series with non commuting parameters
