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Институт Золотого Сечения - Математика Гармонии

Аlexey Stakhov, Boris Rozin
The continuous functions for the Fibonacci
and Lucas p-numbers
Autor info - Аlexey Stakhov
Autor info - Boris Rozin
International Club of the Golden Section,
6 MeCreary Trail, Button, ON, Canada L7E 2C8
Abstract

The new continuous functions for the Fibonacci and Lucas p-numbers using Binet formulas are introduced. The article is of a fundamental interest for Fibonacci numbers theory and theoretical physics.

1. New results in Fibonacci number theory

Modern science, particularly physics [1-13], widely applies the recurring series of the Fibonacci numbers F(n)
(1)

and the recurring series of the Lucas numbers L(n)
(2)

which result from application of the following recurrence relations:
(3)
(4)
(5)
(6)

The ratio of the adjacent numbers in the Fibonacci series (1) and Lucas series (2) tends toward at the irrational numbercalled the Golden Proportion (Golden Mean) [14—16].

In the 19th century the French mathematician Binet devised two remarkable analytical formulas for the Fibonacci and Lucas numbers [14—16]:
(7)
(8)

whereis the Golden Proportion and n = 0, ±1, ±2, ±3...

In recent years, researchers developed further the theory of the Golden Section and the Fibonacci numbers [17-30]. Hyperbolic Fibonacci and Lucas functions [21,24,26], which are extensions of Binet formulas for a continuous domain, have a strategic importance for the development of both mathematics and theoretical physics. In a previous article [28], the authors presented a new function of the second power called the Golden Shofar. The graph of this function reminds one of a horn, which is used to blow in the Yom Kippur (The Judgment Day).

The sinusoidal Fibonacci function [28] is a further development of a continuous approach to the Fibonacci numbers theory begun in a series of papers [21,24,26]. The definition of the function is


Fig. 1. The sinusoidal Fibonacci function.


See full text of this article at PDF format (225Kb).


Аlexey Stakhov, Boris Rozin, The continuous functions for the Fibonacci and Lucas p-numbers // «Академия Тринитаризма», М., Эл № 77-6567, публ.13215, 14.04.2006

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