Abstract
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.
Highlights
During the last centuries a great deal of scientist’s attempts has been made on generalizing the mathematical aspects and their applications
Wider generalization of Binet, Pell and Gazale formulas and wider generalizations of symmetric hyperbolic Fibonacci and Lucas functions presented by Stakhov and Rozin are achieved
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
Summary
During the last centuries a great deal of scientist’s attempts has been made on generalizing the mathematical aspects and their 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.
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