Quantum Calculus of Fibonacci Divisors and Infinite Hierarchy of Bosonic-Fermionic Golden Quantum Oscillators

TL;DR

This paper develops a quantum calculus based on Fibonacci divisors and Golden derivatives, leading to an infinite hierarchy of Golden quantum oscillators, coherent states, and analytic functions with applications in quantum deformation and computations.

Contribution

It introduces Fibonacci divisors and Golden derivatives to construct a new quantum calculus and hierarchy of Golden quantum oscillators with novel algebraic and analytical structures.

Findings

Hierarchy of Golden quantum oscillators with Fibonacci-based spectra

Golden deformed bosonic and fermionic oscillators derived from Fibonacci divisors

Reduction to classical functions as k approaches zero

Abstract

Starting from divisibility problem for Fibonacci numbers we introduce Fibonacci divisors, related hierarchy of Golden derivatives in powers of the Golden Ratio and develop corresponding quantum calculus. By this calculus, the infinite hierarchy of Golden quantum oscillators with integer spectrum determined by Fibonacci divisors, the hierarchy of Golden coherent states and related Fock-Bargman representations in space of complex analytic functions are derived. It is shown that Fibonacci divisors with even and odd $k$ describe Golden deformed bosonic and fermionic quantum oscillators, correspondingly. By the set of translation operators we find the hierarchy of Golden binomials and related Golden analytic functions, conjugate to Fibonacci number $F_k$. In the limit k -> 0, Golden analytic functions reduce to classical holomorphic functions and quantum calculus of Fibonacci divisors to the usual one. Several applications of the calculus to quantum deformation of bosonic and fermionic oscillator algebras, R-matrices, hydrodynamic images and quantum computations are discussed.