A new class of deformed special functions from quantum homogeneous spaces

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