Multi Parameters Golden Ratio and Some Applications

The present paper is devoted to the generalized multi parameters golden ratio. Variety of features like two-dimensional continued fractions, and conjectures on geometrical properties concerning to this subject are also presented. Wider generalization of Binet, Pell and Gazale formulas and wider generalizations of symmetric hyperbolic Fibonacci and Lucas functions presented by Stakhov and Rozin are also achieved. Geometrical applications such as applications in angle trisection and easy drawing of every regular polygons are developed. As a special case, some famous identities like Cassini’s, Askey’s are derived and presented, and also a new class of multi parameters hyperbolic functions and their properties are introduced, finally a generalized Q-matrix called Gn-matrix of order n being a generating matrix for the generalized Fibonacci numbers of order n and its inverse are created. The corresponding code matrix will prevent the attack to the data based on previous matrix.


Introduction
During the last centuries a great deal of scientist's attempts has been made on generalizing the mathematical aspects and their applications.One of the famous aspects in mathematics is called golden ratio or golden proportion has been studied by many authors for generalization from different points of view.In recent years there were a huge interest of modern science in the application of the Golden section and Fibonacci numbers in different area of engineering and science.An extensive bibliography of activities can be found in a paper by Stakhov [1,2] who has investigated the generalized principle of Golden section and its vast area of applications.The main goal of his attention is to state the fundamental elements of this subject i.e. description of its basic concepts and theories and discussion on its applications in modern science.In this connection he explains the idea of the creation of a new mathematical direction called mathematics of Harmony that moved to intend for the mathematical simulation of those phenomena and processes of objective world for which Fibonacci numbers and Golden Section are their objective essence which can influence on the other areas of human culture.Stakhov considers the harmony mathematics from sacral geometry point of view and its applications in this field [3].Some authors consider the extension of the Fibonacci numbers to create a generalized matrix with various properties suitable for the coding theory [4,5].Even they established generalized relations among the code matrix elements for all values of Fibonacci p-numbers with a high ability.Order-m generalized of Fibonacci p-numbers for a matrix representation gives some identities [6] useful for further application in theoretical physics and Metaphysics [7][8][9][10].Some authors determine certain matrices whose parameters generate the Lucas p-numbers and their sums creates the continues functions for Fibonacci and Lucas p-numbers which is considered in [11][12][13][14] and their generalized polynomials and properties are also introduced in [15], based on Golden section the attempt of build up the fundamental of a new kind of mathematical direction is addressed in [3] which is the requirements of modern natural science and art and engineering, these mathematical theories are the source of many new ideas in mathematics, philosophy, botanic and biology, electrical and computer science and engineering, communication system, mathematical education, theoretical physics and particle physics [16].
Other area of applications returns to the connection between the Fibonacci sequence and Kalman filter, by exploiting the duality principle control [17], finally numerical results of generalized random Fibonacci se-quences which are stochastic versions of classical Fibonacci sequences are also obtained [18].Recently Eduardo Soroko developed very original approach to structural harmony of system [19], using generalized Golden sections.He claims the Generalized Golden sections are invariant which allow natural systems in process of their self-organization to find the harmonic structure, stationary regime of their existence, structural and functional stability.
Following the introduction, in Section 2 we establish multi parameters Golden Ratio and geometric applications.In Section 3 we generalized the Gazale formula.Sections 4 and 5 investigate the generalized hyperbolic functions and double parameters matrices and applications.

Preliminaries
In this paper we start a geometrical discussion which leads to a generalized form of golden ratio with multi parameters that for the case of single parameter it changes to the ordinary generalized form which has been considered by authors in the recent years.
Suppose a line AB is to divided in two parts AC and CB (Figure 1) where and a  are real positive numbers.Let where a b   .In a special case for we intend to divide We call the positive root of this equation the general- which is called generalized golden ratio and has been considered in recent years by some authors, for a = 1 gives the famous historical golden ratio where the ratio of the adjacent numbers in Fibonacci series and Lucas series tends towards this irrational number.
Let us consider the properties of this golden ratio, for a = b we have which is one solution of equation .Now other properties of the ratio can be considered.We have the generalized golden ratio  when a = 1 reduces to the well-known historical golden ratio that can be expanded into this fraction or nested roots equations that goes on for ever and called continued fraction or nested roots   Using the  relation in term of itself we get the following continued fractions for  , a  , respectively Stakhov and Rozin [20,21] give some interesting results for and , similar results are verified for different values of a and b, two dimension periodic continued fractions are considered by some authors [22,23] using above formulas may generalize the application in art and architecture.The above line AB can be divided in n sections and by the successive ratios a system of equations will be created such that for a given value n, (1) be extended to multi parameters ratios to create a multi parameters Golden ratio.In this paper we concentrate on a generalized double parameters and applications of this ratio.
, the generalized single parameter ratio different from   which can be generalized to a multi parameters cases. , For the triangles ADC and ADB we have respectively If AB AC  then the ABC will be an isosceles triangle and , so we will have and triangle ABC a will be isosceles and we conclude easily DC  , 1 The impossibility proofs are so advanced, that many people flatly refuse to accept the problem are impossible [24].The proofs of impossibilities of certain geometric constructions like doubling the cube and squaring the circle, trisecting the angle is impossible and details can be found in some references [25], trisecting the angle is impossible that is there exist an angle that cannot be trisected with a straightedge and compass.On the other hand there are some angles which may be trisected, some author give valuable discussion for ideal and classical trisection of an arbitrary angle [26].Using generalized golden ratio one may establish the following criteria for trisection of a given angle.Now, we construct an isosceles triangle ADB whose vertex angle is  ADB   (Figure 3), hence the base .By a similar way we divide in generalized golden ratio and find such that , now the line will contact with the circle in a point say , joining we have having  , a can easily be calculated, for given  when the segments are and a b  respectively, and are given in the Table 1 for and some nominal values of 1 b   .

Generalized Gazale Formula
, , To attain the solution, using characteristics method gives the following quadratic equation where is the Generalized Golden Ratio and Hence the general solution of (1) can be written as Looking at the two initial values 0 and 1 , the coefficients and can be determined, thus For , , which is the Gazale formula for the Generalized Fibonacci sequence, and if n F is the ordinary Fibonacci sequence, the general solution to for the case 1 a  , and for the case gives the John Pell (1610-1685) formula, and this formula generates an infinite number of generalized Fibonacci number of different orders for a and b.
The following Theorem is a generalized aspect of theorems that have been established for Fibonacci and Lucas numbers.
Theorem 3.1 If and are real positive numbers, then and is independent of .a Proof We have , Similar identities like the Askey's and Catelan's and d'Ocagne's can be generalized.For instance in Osler and Hilburn [2] Askcy's hyperbolic sine identity for ordinary Fibonacci sequence is given as and generalized form in [27,28] for real positive is If we generalize (11) for where and

Golden Hyperbolic Functions
Similar to the Stakhov and Rozin hyperbolic functions [20], we can use the same approach and introduce the generalized hyperbolic functions of order (a,b) in the following form Then we will have We define and .
The symmetric property is also satisfied, as we have for Fibonacci and Lucas hyperbolics sine and cosine.For example Similar to the De'Moivre formulas for the double parameter generalized hyperbolic functions following relation is satisfied

Generalized Double Parameter Matrices of n
cci numbers by the fol-lowing relation Hoggatt [3] introduced the following Q-matrix and Theory of Fibonacci Q-matrix where n Q connects to the Fibona     The matrix of order is then for any the procedure can be rameter extended to the case of multi pas, and we guess  and also it maybe generalized to the multi dimension case which allows developing the application to the communication engineering specially to the coding theory [4,5] The inverse can also be calculated similar to the by induction.

Figure 1 .
Figure 1.Dividing the line AB in generalized Golden section.
, actually (2) can be considered as a characteristic function of a difference equation of order n with parameters and b a

Figure 2 .
Figure 2. Dividing BD by point C in the generalized golden ratio.

Let and b
be any real positive number, and define a Generalized Gazale sequence n a he generalized golden ratio is extended to ,

Table 1 . Values of Cos (α), a and  for the nominal values of α.
AM