One-parameter Darboux-deformed Fibonacci numbers
H.C. Rosu, S.C. Mancas
TL;DR
This paper introduces a one-parameter Darboux deformation method applied to continuous Fibonacci sequences, resulting in new deformed Fibonacci numbers and analyzing their properties through Ermakov-Lewis invariants.
Contribution
It presents a novel application of Darboux deformations to continuous Fibonacci sequences and explores their invariants, extending previous work on related differential equations.
Findings
Darboux deformations produce new non-integer Fibonacci-like numbers.
The method links Fibonacci sequences to parametric oscillator equations.
Ermakov-Lewis invariants are derived for the deformed sequences.
Abstract
One-parameter Darboux deformations are effected for the simple 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 (non integer) numbers. Considering the same ODE as a parametric oscillator equation, the Ermakov-Lewis invariants for these sequences are also discussed.
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Taxonomy
TopicsAdvanced Mathematical Theories and Applications · Fractal and DNA sequence analysis · Algebraic structures and combinatorial models