On rational functions with golden ratio as fixed point.

摘要

The existence of an equivalence subset of rational functions with Fibonacci numbers as coefficients and the Golden Ratio as fixed point is proven. The proof is based on two theorems establishing basic relationships underlying the Fibonacci Sequence, Pascal's Triangle and the Golden Ratio. Equations from the two theorems are related to each other and seen to generate the equivalence subset of rational functions. Proof by induction on these equations constitutes the proof of the existence of this subset of rational functions. It is found that this subset of rational functions possesses interesting mathematical properties, particularly that of convergence to the Golden Ratio at the limit. Further investigation showed that this subset of rational functions possesses algebraic structures that would take us into the realms of abstract algebra and complex analysis. The study concludes that the findings are significant as an addition to mathematical knowledge, and as a possible tool for biological research. In this respect, recommendations are made for further research with a view to applications in the sciences and education.