turkmath.org

 Starting from divisibility problem for Fibonacci numbers, we introduce integer Fibonacci divisors, conjugate to $F_k$, related hierarchy of Golden derivatives in powers of the Golden Ratio and develop corresponding quantum calculus. By the set of translation operators, we find the hierarchy of Golden binomials and related Golden analytic functions. In the limit $k \to 0,$ these functions reduce to classical holomorphic functions and quantum calculus of Fibonacci divisors to the usual one. The hierarchy of Golden periodic functions appearing in this calculus, we relate with classical method of images in planar hydrodynamics (electrostatics). Several applications of the calculus to quantum deformed algebras and quantum computation and information theory are discussed. In particular, we show that for repeated consecutive duplicated qubit states, probabilities are determined by Fibonacci numbers. We generalize these results for direct product of multiple qubit states and arbitrary position of repeated states. The calculations are based on structure of Fibonacci trees in space of qubit states, growing in the left and in the right directions, number of branches and allowed paths on the trees.