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The set of mod $n$ functions associated with primitive roots of unity anddiscrete Fourier transform is introduced. These functions naturally appear indescription of superposition of coherent states related with regular polygon,which we call kaleidoscope of quantum coherent states. Displacement operatorsfor kaleidoscope states are obtained by mod $n$ exponential functions withoperator argument and non-commutative addition formulas. Normalizationconstants, average number of photons, Heinsenberg uncertainty relations andcoordinate representation of wave functions with mod n symmetry are expressedin a compact form by these functions.

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