  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.
