One-parameter Darboux-deformed Fibonacci numbers

Abstract

One-parameter Darboux deformations are established for the simple ordinary differential equation (ODE) satisfied by the continuous generalizations of the Fibonacci sequence recently discussed by Faraoni and Atieh [Symmetry 13, 200 (2021)], who promoted a formal analogy with the Friedmann equation in the FLRW homogeneous cosmology. The method allows the introduction of deformations of the continuous Fibonacci sequences, hence of Darboux-deformed Fibonacci (noninteger) numbers. Considering the same ODE as a parametric oscillator equation, the Ermakov–Lewis invariants for these sequences are also discussed.