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

## TL;DR

This paper introduces mod n functions linked to roots of unity and Fourier transforms, used to describe kaleidoscope of quantum coherent states with specific symmetry properties, and derives related physical and mathematical characteristics.

## Contribution

It presents a novel class of mod n functions and applies them to analyze superpositions of quantum coherent states with polygonal symmetry.

## Key findings

- Derived displacement operators for kaleidoscope states.
- Expressed normalization, photon number, and uncertainty relations using mod n functions.
- Provided coordinate representations of wave functions with mod n symmetry.

## Abstract

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.