2020/10/20 by Oktay K. Pashaev
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Theories and Applications #Algebraic structures and combinatorial models #Classical limit #Differential calculus #Divisibility rule #Fibonacci number #Golden ratio #Hierarchy #Quantum #Quantum algorithm #math-ph #math.MP #math.QA #quant-ph
paper · pdf · doi:10.1142/s0219887821500754
published as International Journal of Geometric Methods in Modern Physics Vol. 18, No. 5 (2021) 2150075 (32 pages) · 39 pages
arxiv created 2020/10/20 · openalex created_date 2020/10/29 · openalex publication_date 2021/03/05 · arxiv updated 2021/06/15 · openalex updated_date 2026/08/05
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 [Formula: see text] 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 [Formula: see text]. In the limit [Formula: see text], 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, [Formula: see text]-matrices, geometry of hydrodynamic images and quantum computations are discussed.