Special functions with mod n symmetry and kaleidoscope of quantum coherent states

摘要: The set of mod $n$ functions associated with primitive roots unity and discrete Fourier transform is introduced. These naturally appear in description superposition coherent states related regular polygon, which we call kaleidoscope quantum states. Displacement operators for are obtained by exponential operator argument non-commutative addition formulas. Normalization constants, average number photons, Heinsenberg uncertainty relations coordinate representation wave n symmetry expressed a compact form these functions.