Институт Золотого Сечения - Математика Гармонии

Аlexey P. Stakhov, B. Rosin
Theory of Binet formulas for Fibonacci and Lucas p-numbers
The International Club of the Golden Section, 6 McCreary Trail, Bolton, ON, L7E2C8 Canada
Abstract

Modern natural science requires the development of new mathematical apparatus. The generalized Fibonacci numbers or Fibonacci p-numbers (p = 0,1,2,3,...), which appear in the "diagonal sums" of Pascal's triangle and are assigned in the recurrent form, are a new mathematical discovery. The purpose of the present article is to derive analytical formulas for the Fibonacci p-numbers. We show that these formulas are similar to the Binet formulas for the classical Fibonacci numbers. Moreover, in this article, there is derived one more class of the recurrent sequences, which is defined to be a generalization of the Lucas numbers (Lucas p-numbers).

1. Introduction

As is well known, the Golden Proportion plays an increasingly important role in modern physical research

[1-14]. A substantial number of researchers from various areas of science are inclined to believe that the Golden Proportion is one of the fundamental constants of the "physical world." As early as in Johannes Kepler's research the Golden Proportion was named as one of two treasures of geometry and compared it to Pythagorean Theorem. The outstanding American theoretical physicist, Richard Feynman (1918-1988), who is one of the founders of the quantum electrodynamics, expressed his admiration of the Golden Proportion in the following words: "What miracles exist in mathematics. According to my theory, the Golden Proportion of the ancient Greeks gives the minimal power condition of the butadiene molecule."

The generalized Fibonacci or Fibonacci p-numbers  are one of the important mathematical discoveries of the modern Golden Section and Fibonacci numbers theory [15-34]. Let us define the basic recurrence relation for Fibonacci p-numbers. For any given p (p = 0,1,2,3,...) they are given by the following recurrent relation: (1)

Eq. (1) is then the basic recurrence relation.

It is necessary to note, that for various initial conditions (2)

where a1,a2,…ap=1 are elements of the set of integers, real or complex numbers, we will obtain from (1) the infinite set of recurrent numerical sequences that relate to the class of the recurrent Fibonacci p-series. In particular, if we take (3)

then for these initial conditions, the basic recurrence relation (1) "generates" a class of the Fibonacci p-numbers that are "diagonal sums" of Pascal's triangle .

For different p, the basic recurrence relation (1) "generates" a number of the remarkable numerical sequences that are widely used in mathematics. For example, for the case p = 0, the recurrence relation (1) is reduced to the following: (4)

which generates the sequence of the powers of two: 1,2,4,8,16,32,..., for the given initial condition (5)

For the case p = 1, the basic recurrence relation (1) takes the following form: (6)

This recurrence relation for the initial conditions: (7)

generates the classical Fibonacci numbers F(n) = {1,1,2,3,5,8,13,21...}. Furthermore, given the initial conditions: (8)

the relation (6) generates the classical Lucas numbers L1(n) = {1,3,4,7,11,18,29,...}.

It is known that the limit of the ratio of two adjacent Fibonacci numbers F1(n) (as well as the adjacent Lucas numbers L1(n) and the adjacent numbers of any numerical sequence that is given by the recurrence relation (6)) tends to the Golden Proportion, i.e.: (9)

The Golden Proportion x is a positive root of the following characteristic equation: (10)

that is also called the Golden Section equation. Eq. (10) has two real roots: (11)

Binet formulas are well known in the Fibonacci numbers theory [15-17]. These formulas allow all Fibonacci numbers F1(n) and Lucas numbers L1(n) to be represented by the roots x1 and x2 of Eq. (10): (12) (13)

where n = 0,±l,±2,±3,....

Note that the recurrence relations (4), (6), Eq. (10), and the Binet formulas (12) and (13) are widely used for simulation of various physical and biological phenomena. In particular, they are used for the process of cell division [32,33] and in the description of Fibonacci's lattices on the surface of the "phyllotaxis" objects .

In recent years it has been shown that the Fibonacci p-numbers, given by (1), can be used for simulation of biological cell division [31,33] and the self-organizing systems . Moreover, the connection of the Fibonacci p-numbers to Pascal's triangle has a special interest. It became a source of new mathematical and even philosophical discoveries based on the Fibonacci p-numbers, i.e. the "Law of structural harmony of systems" , the Generalized Principle of the Golden Section  and others. 