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